Scarlet Witch allows her once per game to nullify a team ability on a roll of 1-5. Now my question is ... The abilities listed as team base abilities... Can I nullify team base abilities with her trait?

No, the combat abilities granted by being a team base are combat abilities, not team abilities.

I will hijack this thread to add another AvX Switch question. Can a figure brought in through battlefield promotion use a team ability that Scarlet Witch nullified earlier in the game? Would they have to roll a d6 at that point? Or no?

Since replacement characters are considered the same character, I would think that they would not be able to use it if they already rolled a 1-5 for that ability, but if they rolled a 6 for that, or didn't have to roll at all, they could use it just fine.

You definitely wouldn't roll when you replace the character. You only do it when Scarlet Witch is given the DPA, and that's it.

Here's my example. You have a team of Phoenix Force Cyclops, Phoenix Force Magik, M10 Marvel Girl. Scarlet Witch nullifies Power Cosmic. Marvel Girl Doesn't have Power Cosmic, but later, when she Battlefield Promotes into Dark Phoenix, she does. So at that point later in the game, she gets to use that team ability, correct?

Sounds right to me.