I'm not going to pass judgement on the "dating a married man" situation. We all have our opinions on this. OP is going to do what she wants in that respect.
My sister dates married men. She's looking for a long term relationship and meets guys online who say they are also looking for a long term relationship. She hasn't dated anyone married who hasn't been up-front about being married. The stories why they are dating while still married vary. While none of them have left their wives for her, one thing she has learned is how widely the separation/divorce laws vary from state to state, and she's shared some of that with me in the past. Knowing this, I have an opinion about the pictures, the credit card, etc. For example:
>In some states, infidelity IS still grounds in a divorce suit, and the wife will make out better, financially, if she can show that the husband has a girlfriend. Maybe that's part of the PI-picture taking issue. Doesn't matter if it's innocent or not. Doesn't matter if OP is totally independent and not looking for a roommate or help on the rent or not. All that matters is what it looks like in court, what the wife's lawyer can make it appear to be.
>In some states, the husband can't take his name off accounts without it appearing to be abandonment and similar to stealing. I would guess that both husband and wife originally had their names on the credit card accounts, and the husband has been advised not to cut her off from the card. It's easy enough for the wife to get copies of the bill, online; her having access to it would have nothing to do with where the credit card bill is sent, if she is still on the account (which she may be, based on advice given by the husband's/boyfriend's lawyer).
>In some states, the length of the marriage plays into how things like mutual finances, retirement accounts etc. are divided. Might explain why she is dragging her feet on getting divorced.
I am not saying that any of this is what is or isn't happening, just that, depending on the state, the divorce laws differ and it could be to the wife's financial benefit, in the long run, to build a case agains the husband/boyfriend. Also... I don't blame her for dragging her heels, if she doesn't want to get/be divorced. She might be holding onto this marriage, this financial standard, for as long as possible if she doesn't want to be divorced.