A Few General Questions

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(1). Second Wind from PGL, Bloodbath from MRD, and Invigorate from LNC seem to be popular. The first two restore HP, the last one gives a TP regen buff for a small HP cost. You can find the levels of these skills on http://ffxiv.yg.com.

(2). Sentinel is great because it seems to just cut all damage to 1/3rd for its duration, which makes it extremely powerful. E.g. on THM if things go sour and we pull too many, I can throw out Shadowsear, pop Sentinel to tank the damage, and then I can survive long enough to throw out a Drain II. This (hopefully) is enough damage to kill most of the enemies, esp if the rest of the party is continuing to fight. The tier 2 versions of the abilities from (1) are also quite high-level and useful.

(3). I prefer THM for damage and enfeebling, but unless you want to play a caster many of its abilities aren't as useful cross-class. CNJ, on the other hand, has Chainspell (at 44), which will let you Raise two people at once, one nearly instantly. It also has a more helpful array of buffs.

(4). I have CNJ and THM at 50, but my next highest is GLA at only 36. It seems like I contribute plenty in Dungeons, NM/Boss Fights, and Strongholds. In fact, I'm holding off on taking other things too high because I want to have lower-level classes to play later content. I missed out on Shposhae because I have nothing below L25.

(5). [Bah! I misread your original question! This refers to DoW/M. DoL or DoH can take quite a while, but I can't provide solid numbers as I rarely do grinding for DoLs or DoHs.]
With parties or powerlevelling I've heard of people doing it in a week full-time. A more normal timeframe is one to two months, especially if it's your first class. Leves and parties, esp. 20+, are the way to go.

(6). You can focus on a single DoL, though traits from other DoLs can be helpful, they aren't as important. Cross-class abilities are a little more important for DoHs, and most DoHs get a generally useful ability at L10, so many crafters have all of them at 10.

(7). Yes. However, leves are great for any class due to the XP bonus for successful completion. Keep in mind for any battlecraft leves that it's better to finish quickly to secure the speed bonus than to eke out a bit more XP by fighting stronger foes but resting between fights.

(8). Preserve (ALC10) and Fulfillment (LTW10) are always on my bar, regardless of the DoH class.

(9). No idea. I'm on Wutai. I know Figaro has a big NA Linkshell due to reddit.

(10). No. Join the Maelstrom because it's awesome and our leader has a friggin' gun. It honestly doesn't make that big a difference. Join the one you like best.

(11). It's really hard to tell right now because the market hasn't settled from the upset in 1.19. Finished products generally sell well at the right price unless there's a glut of them, which you can check from the Item Search Counter. AFAIK there is no such utility online, but maybe the new price history feature will have date/times.

(12). Yeah. They're fixing this in 1.20. The server program they inherited from the original development process has issues. It sounds like you're making the best use of the system. By the way, if it's not available, but you want it, there's probably a market there!

Hope this helps!