Zophos

First (because I don't think anyone has answered this part of your question): Yes, you will be able to EVA the stranded Kerbal across to your rescue ship. In fact, you can't generally dock with the stranded parts because they don't come equipped with docking ports.  When you get the grabber, you can grab those parts if you really want to avoid EVA.
As for the rendezvous, there are a few points worth mentioning, some of which have already been addressed:
If you put yourself in a lower orbit than the target, you will "gain" on it (travel a larger angle in a given time). A higher orbit is the opposite. The larger the difference in altitude, the faster your angle will change relative to the target. Either way will work; if you are close behind your target and in a higher orbit, it will get farther and farther ahead, until eventually it is coming up behind you. Of course, this may take a few orbits. Crossing the target's orbit (altitude) is inefficient, as you will spend part of the orbit gaining and part losing relative to the target. So pick one. If the target is in a low orbit, you have much more maneuvering room if you pick a higher orbit and let it gain on you. With all that in mind, what I usually do is launch to a circular orbit at about the altitude of the target's apoapsis. Then, at that apoapsis (wherever it falls on my orbit), I will burn prograde to raise my apoapsis some way above the target's orbit.  This puts me in an elliptical orbit higher than the target, so my orbit will take longer than the target's (it will gain on me). Moreover, my periapsis is at the same point as the target's apoapsis, and it will stay that way as long as I burn only at my periapsis (raising or lowering my apoapsis). 
My goal at that point is just to make sure I hit periapsis at the same time the target hits apoapsis.  Generally, only a handful of orbits is needed to get one where the target will be relatively close behind me at the meeting point --- specifically, where it would be ahead of me on the next orbit if I don't do anything. At that point, I burn retrograde at periapsis to lower my apoapsis; not all the way to match orbits, but just enough to let the target catch up to an extremely close encounter at the next meeting. Finally, at the encounter, a second retrograde burn matches orbits (lowering my apoapsis to match the target's periapsis).
You can apply the same ideas whether you choose to meet at the apoapsis or periapsis of the target's orbit. But by staying strictly above the target, you will never hit atmosphere as you've described.
There are also more sophisticated techniques in which you enter a circular "phasing orbit" and then do a Hohmann transfer to the target at the right point. They are more efficient in some cases, but they involve more burns and more calculation.  The technique described above can be done entirely on the screen with maneuver nodes and your eyeball.

In the hopes that this will help the console crowd, I'll show the early-career tail-dragger biplane I put together for a Redditor yesterday. Nothing above the 45-science tier, no tweaks to anything except control surfaces, and it takes off from the grass with no control inputs other than starting the engines (at least in 1.1.3 on the PC).  It can also hit 20km altitude (once) to complete survey contracts. Some construction notes in the album:

And the takeoff roll (EDIT: on grass, to avoid using the hilariously bad starting runway). The only control inputs come from SAS (which keeps the nose up).