Sport is the base level trim.
Sahara is the luxury trim. More likely to have leather seats and usually had a same color hardtop. Same offroading as sport, other than the fact they tended to come with 18" wheels.
Rubicon is the offroader. Locking difs and 4.10 axles ratio. Also had beefier axles.
For what you are looking to do, I would just keep the box for manual transmission and two door checked and keep and eye on what shows up.
The closer to 2018 you get the more refined the jeep, though anything 2015 and up is just fine. 14s are usually just fine as well, A 13 that is still on the road is probably ok, but I would stay away from 2012 model year. It was the first year for the 3.6 engine and as most first years, there were some issues.
if you are looking for a bargain JK, the ones with a 3.8 are generally fine as long as you keep an eye on your oil. Just never get a 3.8 with an auto trans.
I have a 99 TJ and a 2009 JKU Rubicon. The TJ is a blast to drive, but I wouldn't drive it across state without a good deal of rest before hand. I daily the JK and take it across state to families quite a bit. I'm about to hit 211K miles.