Help me pick a new character

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Just as a note, the explorer classes were pretty significantly revamped in the RED patch, so a few of them are much different than what they used to be. Just pointing that out in case you're avoiding them since you already tried the old versions.

1. I have to disagree with Phantom and agree with the Xenon choice by other posters. In particular, Phantom isn't terribly great in terms of powerleveling or raw DPS in my experience, even if its link skill is pretty good. Xenon seems to fit your requirements the best:
a) For powerleveling, 2nd/3rd job is average, 4th job is amazingly fast.
b) The link skill for Xenon is %stat, which will be valuable for any next character you make. Also, being good with bossing means you could potentially sell boss drops to fund future characters.
c) Xenon's raw DPS is pretty decent and scales up very well with equips.
d) Xenon is quite mobile and is good at both bossing and mobbing
e) Xenon has some laser-like skills

2) The only thing that I have to add to the previous posts is that grouping character cards by class archetype (warrior, mage, archer, etc) will give you additional bonuses.

3) You can only have one of three professions: Alchemy, Smithing, and Accessory Crafting. Most of the things that you can make aren't game-breaking except for the newest update to professions, where you can get to level 11-12 (called Master Craftsman and Meister level) in professions (the normal max is level 10). You need to craft items daily to stay at these levels, and you can create cubes with Smithing at those levels and pretty good potions with Alchemy. Accessory Crafting doesn't have as interesting stuff at Meister level, but you can easily create Half Earrings which you can also fuse to get some decent potentials off the bat for a small funding boost.

4) Can't comment on this

5) Besides what other people mentioned, you should make sure to farm the upcoming Valentine's Candy event starting on 2/07

6) Ghost Ship, Ninja Castle, LHC, Neo City give you mastery books on completion. Not super valuable but will save you a tiny bit of mesos.
Definitely do Ghost Ship for the badge also.
Gollux and Commerci are quite good for higher levels (140+) as far as bossing.
Can't think of any quests off the top of my head.

7) Not really. It's the same as always: either get valuable drops from highest-tier bosses or merch your way up.

8) A little too vague for me to think of anything in particular right now.

EDIT: The only reason why anyone would suggest making Mihile first is to get the stance link skill. Otherwise, that class is a total waste of time (imo).