1) Other than the few quests you get in the beginning that get you acquainted with your starting area, no...not really. Overall the story line is relatively the same across the board with starting areas, class, and/or race having little to no impact, what-so-ever. Race in particular means precisely zip regards to story.

2) I'm...not really sure I've ever noticed any particular lacking or over-abundance of any specific race.

3) I recently hit level 50 for the first time with my Ninja job (after basically switching out from 45 SMN), with varied lvl alt jobs. My gathering/crafting jobs however...i have basically left untouched to date. Think i may have weaver at like...3 and botanist is sitting at like level 8 or 9. So no...they're not specifically required to do or anything (for the most part, anyway*) and can be delayed if you wish.

*Case in point. As mentioned...I just hit 50 with Ninja. I've not had to do a single craft or gather for the entire trip (nor at any time for any of my other classes/jobs). Then I went to my relic weapon quest and discovered that...to make the knives I need a lvl 50 alchemist. Well THAT sucks. That being said...more often than not...should you need a crafter for something, there is not shortage of folks that are willing to put their skills to use for you (especially if you're providing the materials...though some may or may not charge you for services rendered. I got lucky and found someone willing to do it for free.)