1. There are a lot of side quests that are missable, but I would say that a lot of the real important ones can be done anytime after the last scene.
2. Yes... Or at least I missed a bunch my first time.
3. None that I can think of. What I did was beat the game without going out of my way looking up side quests, then on my second playthrough I looked up a guide and got most of the stuff I missed. Actually, this isn't missable, but make sure you do the Sheridan crafting side quest... It doesn't matter which one you choose on the last stage, but remember which one you did so you can make the other on your next playthrough so you can get both in your collectors book.
Oh, and try to get the Berserker title for Luke by fighting 256 battles on hard. I hear a lot of people have trouble getting it in New Game+.. --- "I am the bringer of the end, fear me, I am the beast that is technology."
Cyllya's site should help out quite a bit. As far as quests.. not really.. If you want to start on some quests.. Look into the Blacksmith Side quest and collect Gold Dust and Rocks from the enemies in Zao. For face-chats, there are a few dozen, you can look into Kouli's guide about it.
If you haven't gotten the Winged Boots, in Chesedonia, you get a second chance (also in Kouli's guide, though I doubt you don't have them)
A few quests to remember are any that have an Akzeriuth cut off like the Sword Dancer series, Sonic Thrust (sleep in a St.Binah,Kaitzur/Port Inn)
I guess if you really want to go a bit all out, by this time if you're really going nuts with Din's Trade Shop, you could have gotten Last Fencer, Swordian Dylmos, Star Mace, Full Metal Bow, Penetrator, a few accessories like Mystic Mark/Symbol, and a few of Anise's Dolls. That is assuming you collected the necessary items: http://www.gamefaqs.com/console/ps2/file/929343/45708
However it kind of kills the fun and difficulty in batting, It does however allow you to get higher grade I guess.. since you clear battles in short times.. It's up to you. It's not a side quest, but you could get the Curious Princess title through Din.
1) Yes. You can miss a lot of stuff. 2) Yes if you aren't using a guide. No if you are. If you don't want to miss a lot, use either one of the full walkthroughs or the Extras guide. 3) Yes. Some sidequests (or part of a sidequest) are open for short period of time.