The vice cop takes bribes, overlooks crimes, gambles illegally, and is just generally a dick, yet he's a rockstar (no pun intended) in the department.
The vice cop is a "rockstar" because of his charisma, not his underhandedness.

Affable Cole, who everyone says is a great police officer, has a tryst with a lounge singer and he is persona non grata.
Big stigma in the timeframe, I think.

why is Cole cheating on his wife
I felt it was out of character for him, but it is explained by his post-war stress and that the tortured lounge singer is the only female he knows who can relate; he even mentions in DLC he's a sucker for blondes in casual chat with his partner. Even through all of this, I felt Cole betrayed his values and me by doing such a thing.

I just assumed he wasn't with her romantically (aka my headcanon) and he was framed for a relationship when it was a friendship.

But this doesn't explain why he couldn't confront his wife proper.