I'm a certified 12v installer with 7 years of experience behind all types of remote start systems and installations on manual transmissions. I've only ever driven manual transmissions (except my Dakota) and they all had remote start systems. If you pick a reliable remote start from a reputable supplier (Viper, Code Alarm, Ungo) then you should have no problem getting it installed on your manual Jeep. Don't get it installed at a future shop/best buy/etc type store. Go to a dedicated 12v installation centre and make sure they are trained by government programs, not store training. That will guarantee that you have it installed properly. Most systems have a safety makes you do a string of actions before turning the key off (car out of gear, feet off all pedals, pull E brake, press required key on remote, turn key to off, exit vehicle and system will arm when the door is closed). Opening the door or skipping a step will cancel the setup. If you concerned about the door being off, I wouldn't think you'd drive doorless in the winter, and if so, why do you need a remote start if it's warm enough for doors off?
Do yourself a favor and get a combo remote start/alarm. You'll thank yourself when someone tries to get in and steal our car while it's running.
If you have future questions, don't hesitate to PM me.