I'd say the biggest red flag is not being able to trust your husband when it comes to him being out and needing photo proof. It sounds exhausting on both sides if nothing is going on.

If you feel like he's cheating you should have a direct conversation. If you haven't already shared your full worries with your husband.

I can see wanting to stay with a close friend for moving so that's not too odd to be honest though I can understand it's frustrating to have someone in your space.

Honestly this is an ESH for me, seems like something is weird but without further proof or a conversation with your partner it also could be you're allowing paranoia or jealousy to drive this further.