I'm going to be honest and say I think that you're being unreasonable, OP.
I do believe in trusting your gut instincts, however, you also have to look and consider the evidence. At the minute, from what I've read, he's started going to the gym and taken up running. This is following comments where he's complained about feeling unhealthy and overweight. So, it makes sense and isn't out of the blue that he would suddenly start working out.
He woke up in the middle of the night and played on his phone as he was struggling to sleep. I do this all of the time. For some reason, a bit of screen time helps me sleep.
He's going to see his family alone when you'd normally go together. Is this really a big deal? Maybe he genuinely is going there-and maybe he just wants some alone time with them or wants to tell them something in private.
Unless he's got form for cheating-or lying, I wouldn't be considering any of this particularly worrying at all. Yes, there's such a thing as trusting your gut and quite often, our gut instincts can be proved correct-but you still have to be able to look at the "evidence" and suss things from that, too.
I know if my DP did all of those things, because I trust him and he has no form for cheating or lying to me, I wouldn't remotely think "cheating."
Maybe you do need to sit down with him and talk about things. If he is indeed cheating, and I hope he's not, you need to find out, and if he's not, you need to work on your trust issues.