With the PTW stacking, do you still need to move each unit to attack or can you attack by simply moving the entire stack?
You still need to attack with each unit. The exception is auto-bombard. Now you can assign artillery to continually bombard a square each round until there are no more targets. (it seems to only work with artillery/bombers, not catapults or cannon, which are next to useless anyway).
Also, what about healing damaged units? You still need to move them individually back to one of your cities correct?
Correct. If all your tanks are damaged but your MI is OK, you can use the 'move stack units of same type' button and move all your tanks to a city.
I never used armies in CIV3. Would that help me? Don't I need a great leader to form an army and there are few of those? Also, your army units cannot be upgraded which seems to be a big waste
Highly debatable. If you are playing a conquest/domination game, and army of cavalry may be very useful to you, but if you're going for space race, you're better off using Great Leaders for rushing wonders/improvements. And yes, once a unit is in an army, it's stuck as that, so you may have an army of knights running around in the modern age. You can disband any army for 100 shields, though.
It can be difficult to avoid wars, but not impossible (many people have strategies, eg give in to demands/make frequent trades). I always have war declared on me, even when I try to avoid them, for one reason or another. You can also fight a defensive war, just build defensive units that you don't have to move, and a few offensive ones to pick off survivors. Then sue for peace when their economy is lacking. Unfortunately, the grind of warfare in the modern era is something that can not be fixed with an add-on (they have done what they can to simplify moves, as noted above). It's more a matter of luck and playstyle.