Zeiss Ikon

Still using Windows  (just by way of asking the questions you wouldn't think to ask)?  If you've switched to Linux recently, you need to use right-Shift in place of Alt.

To expand on @Signo's answer, hot staging is something you do because of ullage.  It's manifestly unnecessary in the stock game -- stock engines and tanks don't care about ullage in any way.  If you're playing Realism Overhaul, or pretty much anything using RealFuels and real world engine parts packs, you'll need to pay attention to ullage.
"What's ullage?"
Glad you asked.  Ullage is the "empty space" in a propellant tank.  There's always some; usually filled with some kind of pressurant (even pump-fed engines need pressure in the tanks, just not as much as pressure-fed engines).  This can be bad if the pressurant winds up near the rear/bottom of the tank, where there should be propellant, when your engine needs fuel (or oxidizer).
Let's say you launch a basic sounding rocket, like a WAC Corporal (1948 technology here, hang onto your hat).  This is a primitive two-stage rocket, with a high thrust Tiny Tim booster (originally an air-to-ground unguided rocket from WWII).  Your basic WAC Corporal stack will pull something like 30 G off the launch pad with this booster -- but the booster burns out after about 0.6 seconds, leaving you falling upward at roundly 200 m/s.
If you now want to stage and ignite your (pressure fed, hypergolic liquids) upper stage sustainer, you'll find it doesn't work.  EVER.  Because the air drag at 200 m/s results in enough deceleration to immediately slosh all the propellants to the nose cone end of the tanks, and the nitrogen that's supposed to push the aniline/furfuryl fuel and red fuming nitric acid oxidizer into the thrust chamber instead runs straight out through the injectors.  This is why ullage has to be managed.  The WAC Corporal method, as of 1948, was to ignite the upper stage at the same time as the Tiny Tim, and time the staging decoupler to fire at the exact same time the Tiny Tim burned out.  That gave the sustainer engine most of a second to come up to thrust, and ensured the booster wasn't hanging on creating drag that could (even with the sustainer fully ignited) lead to pressurant in the feed lines instead of propellants.  Done this way, the WAC Corporal worked about 95% of the time, and would fly as high as about 120 km (less if it was carrying a payload, of course).
And that's hot staging.  As Signo noted, the Soviet space program used hot staging a lot -- it's the reason you see many of their rockets with open strutwork interstage structures, so that the next stage exhaust has an exit to prevent either choking the engine before it's fully ignited, or blowing out the interstage fairing with potential to lead to a structural collapse as maximum G load.  By contrast, the American space program almost never used it (can't say for certain they never did, but I think that's true).  Look at a Saturn V -- specifically at the base of the S-II second stage.  It has (relatively) small solid propellant "ullage rockets" -- there to ensure there's a (small) positive acceleration after the staging event, so the ignition of the next stage can take place successfully.  And there are no ullage rockets on the Service Module, because it was intended to fly in vacuum and needed RCS to point before a burn anyway, and RCS works just fine for ullage management.  I think there are even Apollo recordings where the crew mentions the ullage burn start, several seconds before the burn to return from Lunar orbit to Earth.
Ullage can be managed other ways, of course.  One way (used often for things like attitude jets/RCS and long-term station keeping thrusters) is a bladder tank.  This stores the fuel (usually a monopropellant, most commonly hydrazine or a derivative these days) inside a bladder, much like a water balloon inside the rigid tank structure; pressure is applied to the outside of the bladder, and the inside is carefully debubbled before launch, so that every time you open the valve, the propellant flows into the catalyst pack and turns into thrust.  In the case of Voyager 2, even if the last time was thirty years ago.
If you're launching from the ground (even if that "ground" is Lunar regolith), gravity (even if it's a fraction of a G) does an admirable job of managing your ullage.  I don't know that it's actually been done in a rocket, but another possibility would be an aerospace version of what I learned to call a "clunk tank" when I was flying radio control models (before they were electric).  This had the fuel pickup in a weight on the end of a flexible tube which would always find its way (with an audible "clunk" if you checked it on the ground with the engine stopped) to the current "bottom" of the tank.  Works great when (as with an aerobatic model) you aren't sure where you local gravity will be any given moment; won't work any better than a fixed pickup in microgravity (so probably not very useful in actual rockets).

I'm playing in RSS/RO/RP-1/Principia (and a bunch of other mods pulled in by those or recommended in the Golden Spreadsheet).  In RSS, you need a hybrid of the old-KSP "burn vertically to some altitude, then start tipping" and the current gravity turn, along with some specific guidance late on the way -- because most craft flying to LEO burn continuously to apogee (which becomes perigee after burning some more to raise the perigee above the atmosphere).  The manual pitch program in MechJeb's PEG ascent guidance does that -- burns straight up for a while (recommended, to around 100 m/s), then does a "by the clock" tip-over, which, if set up correctly, will segue into an "unguided gravity turn" if launching to "current inclination" -- that is, launching due east and letting gravity turn your path southward (or northward, if you're at Woomera or other southern hemisphere site).
Regardless, I found the cause and solution of the problem (though I still consider it a bug, in that MechJeb's PEG module doesn't warn for an invalid entry).  With my four stage rocket, I'm not allowed to enter a stage count of "4" in the "Num Stages" field (just above the button to "Reinitialize Stage Analysis") -- apparently, I have to enter "2" there (despite the fact that Orbiter One needs all four stages -- Tiny Tim boosters, LR-79 booster core, 2xAJ10-37 stage, and final 1xAJ10-37, which apparently will have a few tens of m/s to spare at a 230x230 km orbit).  Do that, even after launch, and the vertical ascent countdown picks up counting down from clamp release, manual pitch program engages, and when its countdown elapses, MechJeb transitions to "unguided gravity turn" as it should.
In the process, I also found a setting that lets me shrink MechJeb's UI, so it doesn't cover 60% of my KSP window when I'm flying an orbital ascent...

If you launch one craft with X amount of dV, when you dock it to another, the dV of the pair will be significantly less -- if they have equal mass, it'll be half what it was.  This is because dV is mass dependent -- you can see this in the VAB if you use Mechjeb or Kerbal Engineer; take any craft you like, add some mass, and watch the dV drop.  You'd have to have both craft carrying X dV in order to still have the same amount when you dock.
Of course, this is more complicated than that simple case; if the two craft have engines with different Isp, the same fuel that would give one 8400 m/s might give the other only, say, 6000 -- or might give it 10,000 if it has much more efficient engines that use the same fuel (are you using Real Fuels?).  But bottom line is that dV is impulse divided by mass (impulse is thrust times time).  Your impulse is constant (unless you spend some of it by burning propellants), but the more mass it has to push, the less dV it produces.

If tourists hang around the Astronaut Center or VAB (waiting to be assigned as crew), there's either an uncompleted contract, or a bug.  If the contract they followed in has, in fact, been satisfied, then feel free to call Security (i.e. the cheat menu, Alt-F12) to politely escort them back to their hotel and, if necessary, the airport for the flight home.

If the object has been captured due to gravity assist from the Mun (and everywhere I type "Mun" you can substitute "Minmus", though its much lower gravity means it's less likely to capture objects in the first place), it might well have apoapsis near or even a bit inside the Mun's orbit -- this would seem to be a stable orbit, but in fact, such an orbit will always reenter the Mun's SOI at some point (depending on the orbital period, this time frame might be "next orbit" or "rather a long time"); that next encounter will change the object's orbit.  If it doesn't impact the Mun, the Mun's gravity will either raise the apoapsis (with the likelihood of ejecting it from Kerbin's SOI) or lower it (but never so far it can't reenter the Mun's SOI -- I can't show you the math, but that's just the way it works).  In either case, the periapsis may also change; this means the object may be lost in four ways: impact with the Mun, impact with Minmus due to Munar assist, ejection from Kerbin's SOI due to Munar assist, or impact with Kerbin due to Munar assist (slowing/lowering the orbit).
Bottom line is, any such capture will be temporary.  Note that the Earth has such a captured asteroid orbiting beyond the Moon, but because Earth runs under Principia instead of patched conics, the body in question actually occupies an Earth-resonant Solar orbit, never getting far from Earth, but not really orbiting the Earth, either.  This body is estimated to have been in this kind of orbit for at least three hundred years, and is likely to remain so for some centuries to come -- but will eventually escape into an independent Solar orbit, like that from whence it came.

MechJeb isn't necessarily perfect for any given maneuver, especially if you're trying to optimize for dV.  In general, the method to launch to LKO with minimum dV (as I understand it) is to set up your gravity turn as early as possible, have the correct thrust-to-weight, and burn continuously until your Pe climbs to your Ap height (which means you've circularized).
Normally, however, we approximate this -- for instance, my TWR with my (curently) standard "almost SSTO" rocket is always too high for my gravity turn, requiring a shutdown of nearly two minutes before my circularization burn.  The approximation that reportedly works best is to start your gravity turn below 1 km or 100 m/s, such that you reach 45 degrees from vertical close to 10 km,  have stages with the correct TWR to keep your Ap marker approximately one minute out from the time you pass 15 km altitude until you shutdown to coast to orbital height, then burn again centered on Ap to circularize -- or steer your climb angle to maintain that time to Ap.  I don't attempt to do this, because I normally fly without fins (unnecessary weight and drag if you have gimballed engines), and turning more than a few degrees from prograde above 100 m/s will result in the rocket tumbling.  Throttling to keep from pushing Ap too far out increases gravity losses.
If, OTOH, you do fly with fins, you may find this difficult in that the rocket will self-right to prograde, and it takes a LOT of pitch or yaw authority to overcome this enough to raise or lower your prograde vector enough to matter.
The subway maps are generally based on vacuum Isp, as you surmised; the problem is, different engines have different levels of Isp increase from sea level to vacuum -- unless you're using the same engine that was used to generate the data for the subway map you're using, "your mileage may vary" (to use an old American phrase from car advertising in the early days of EPA fuel economy ratings).
The best I can offer is to tell you that my favorite mid-game lifter, a Twin Boar with a 6400 tank on top, will SSTO with up to about 15 T of payload; adding drop tanks or boosters will increase that figure some (but then it's not a single stage, is it?), and my launches aren't particularly well optimized either.  i don't use MechJeb or KER, so i don't know the exact dV I consume, but I know that this booster and its variants with tanks or boosters will put the same load into LKO, every single time, and as long as my spacecraft doesn't mass more than that figure, it'll get to orbit (with just a little upper stage fuel consumed to finish circularizing, so the booster will reliably fall back to burn up on reentry).