MockKnizzle

...why not just do the easy thing and simply transfer up to 2868.76 from a parking orbit, then circularize?

I believe that's the HOME heatshield, is it not?

Using some simple math and knowledge of Kerbin's rotational speed, radius and mass, you can calculate the necessary altitude for a Kerbostationary orbit (yes, it's possible). All you need to do is get into a parking orbit and then Hohmann transfer up to a higher synchronous one. Alternately, you can just go on the wiki page and look it up. As far as getting the station situated directly over KSC, that's just a matter of transferring at the right time to put your apoapsis at the right longitude, which is easy since you just start your transfer from the directly opposite side of the planet.

The number of engines shouldn't affect your balance if you're mounting them radially, unless you're not using the symmetry snap options for some reason. To check that your engine exhaust isn't hitting the landing legs, just launch your lander without a lift stage and deploy the legs on the pad. Throttle up briefly and you should easily be able to see if you're gonna have interference problems.

Perhaps Kerbals are more radiation-resistant than their pink-skinned Terran analogues. I bet it has something to do with being green. Also... Jool is larger than Saturn...? Wut? Jool's radius is 6000km, while Saturn's is around 58000km.

When transferring to Duna, I usually make all of my corrections as soon as I'm out of Kerbin's SoI. By using MechJeb to align my craft in the cardinal directions, I just use a little trial and error to adjust my intercept. Typically I'll watch the approach distance/periapse and burn prograde/retrograde as necessary until I can't get it any lower, then I'll re-orient the ship radially and do the same thing, and then back to prograde/retrograde and so on until I've gotten as low as I can. If the inclination difference is small (fractions of a degree) then I can adjust it as well, otherwise I'll wait for a convenient node and make another round of corrections there. This lets me get 100km or lower periapses with great consistency, usually with only one set of correction burns right at the start of the trip. It might not be the most efficient, but I try to design my spacecraft with large enough fuel margins that it doesn't really matter. Once I'm within Duna/wherever's SoI, then I'll immediately make any final corrections necessary to set up Ike avoidance/aerobraking/capture.

Coincidentally, I just performed the exact same mission a day ago. As corvus said, you're gonna want to first get into a parking orbit and then initiate your transfer burn on the other side of the planet. While it might theoretically be possible to launch in some way that you raise your initial apoapse to kerbostationary altitude in a single burn, it would likely involve some strange +/- radial aiming to get the longitude correct. I'm unsure if MechJeb 2 has any functionality for this, and I know the previous version doesn't. Easiest to just set it up with maneuver nodes from a parking orbit.