I used to run that setup. You will need either wheel spacers or 17x9 wheels. I never ran into any rubbing. You won't need to regear but your MPG will suffer by about 3-4 MPGs. In my 2010 JKU I went from around 18 to 12. I added a superchips and at the 93 octane tune I'm getting about 16mpg now average.
Incorrect. OP states he has a 2013, which means he has the 3.6L Pentastar. Don't know whether or not he has the manual or automatic trans, but either way it will handle 35's without a regear no problem. He might lose 1-2 mpg, at most, but that is expected when going to a larger tire.
OP...No, you won't need a regear for 35's. Will it help? Yeah, it'll help acceleration a bit, but I wouldn't bother until you're ready to start taking it off-road, as that's where a regear is going to shine.
Along those lines, yes, you can fit 35's with 20" wheels on a 2.5" puck lift (coil spacers) without any issue. People do it all the time. Shouldn't notice any scrubbing as long as you either put wheel spacers on or get wheels with at least 4.5" backspacing.
If you're building a mall crawler, then 20" wheels are at least acceptable, but most people will tell you not to go any larger than 18's on a Jeep (or any other 4x4 for that matter) because they're not really suited for off-road use. And of course if you go with 20" Rockstars you'll never hear the end of it!
Your Jeep, your priorities, but yeah, you'll be fine with a 2.5" puck lift and 35s as long as you have the proper backspacing on your wheels. I run 37's on a 2.5" coil lift without issue.