What about your force?

Example: I have a Braniac copying your Mystics TA
can any of my other Wildcards copy that TA from my siphoner?

In the past it was ruled no, but with the new relaxation on the Wildcard rules, I was wondering if this still held fast.

Player's guide is great resource

SIPHON POWER
 A character with Siphon Power cannot use team abilities that wild cards cannot normally copy.
 If an opposing character is using the Saboteur feat card, the character with Siphon Power may only choose the team ability of the character with Saboteur. If the opposing character with Saboteur had a team ability that wild cards cannot normally copy, the character with Siphon Power will be able to use that team ability.
 A character with Siphon Power may not have the team ability it is using copied by wild card characters.
 Wild cards may not copy the rewritten team ability that Siphon Power grants.

And should probably mention that the Batman Ally Team Ability has nothing to do with keywords. The Batman Ally Team Ability is a standard team ability from the PAC and will be printed on the base.

Question about Taskmaster Specifically, since he is a Thunderbolt and Wildcard, can he get mystics twice with only himself?

No. Thunderbolt only allows to use the chosen TA if the character cannot already, so using the Mystic TA via his wildcard TA will prevent taskmaster from using it via Thunderbolts.

And even if it did work to get it twice, he would get no real benefit from having it twice. That was established back with SR Thor and Loki Tbolting to Mystics.

But just like with T&L if Taskmaster Tbolts Mystics and then Wildcards a Crossgen figure.... He gets double the effect since it is 2 separate team abilities (even if they do the same thing)

Tarnish

Keyword(s): Thunderbolts
Cost:8
Once at the beginning of the game, choose any one Marvel
team ability other than an alternate team ability, a wild
card team ability, or a team ability that can’t be used by
wild cards. Characters using this team ability can use the
chosen team ability if they can’t already.

I think it's just in the Thunderbolts Ability not because of the previous TandL. I see how they made it and after rereading the last line like 10 times it finally clicked.

Yes, they are.

I'm hoping that with the re-release of the F4 Starter, the Rule Book has a definitive statement regarding ATAs and wild cards, as I've had two judges in the past few months rule definitively that alternate team abilities cannot be copied (despite evidence to the contrary).

Not sure how much more definitive you think it can get.

From p. 17 of the Blackest Night Rulebook:

It's the same exact rulebook as the BN starter.