Can Fylers fly right over other characters while indoors? or do they become adjacent?

You can't soar indoors so they would become adjacent.

Don't fliers ignore character bases (except when they start out based)?

If you are asking if flyers can move past (or through the same square of) opposing grounded figures then yes. If they stop next to a grounded opposing figure then they do become adjacent.

Why would some characters have leap/climb and be flyers if that were the case?
I might be wrong here. I'll bow to superior judgement.

They do indeed.

1. So that they can make close combat attacks from soaring against non-soaring figures.
2. So that they can use Pounce.

In Man-Bat's case, it's been errata'd to be L/C, not flight.

In Mimic's case, it allows him to not need to break and attack characters at different elevations while gaining the benefits and disadvantages of being a flier.

It all makes sense now. Thanks guys.