1 - Ability to customize basically anything and everything was the biggest draw for me (Im also a new iPhone user with a 3GS), but I haven't even gotten into customizing stuff yet because I've been discovering all sorts of other great stuff. Like everyone else has said, ability to get non-Apple approved apps is great. Be it things to change settings quicker (SBSettings) or something that let's you download streaming videos from sites like YouTube or others like it (iMobileCinema), the list goes on. Also, since everyone else has either dodged this fact or just forgotten, you can "acquire" most of the apps/games from the Apple App Store for free, and you can also get a program that let's you download torrents just like on a PC, giving you access to movies and music for free, all right on the phone itself. I'm not saying you should do this (or that you shouldn't), nor am I naming the programs that let you do this, because these forums aren't for that. I'm just stating that it's something you CAN do after jailbreaking, should you so choose.
2 - See #1 for my reasons (also check out this thread, the pics in here were the very reason I decided to JB immediately:
https://forums.macrumors.com/threads/605826/ ). As far as the non-Apple programs, pretty much everything that others have listed. I actually haven't dug too deep into Cydia apps yet, but I have found a bunch of cool stuff just from browsing quickly when bored.
3 - Yes, the phone still has 100% "stock" functionality, everything you could do/access before the JB you can still do after. Now you just have the ability to do tons more that you couldn't before. Cydia is just like the App store, just full of all things Apple basically doesn't want us to have lol.