I think there are a huge amount of assumptions here. And so many people know he is cheating and egging on OP to LTB. Some of you may act on this level of suspicion, but this is someone's life and not a soap opera.
I think it's reasonable to suggest that he might be cheating and that they think his excuse is weak, so would be looking at things closely. And he may well be a cheat- I can understand the suspicion. But there's a difference between that and absolute certainly of guilt and telling OP to end a relationship which by her own admission she was "invested in".
That said, I think it is also possible that he innocent- given what OP says about how he works during the day and spends most evenings/weekends with her at home or socialising together. Other than one night with his brother. Of course he could be conducting an affair when he is supposed to be at work... but then the condom was in jeans he doesn't wear to work, so how does that fit? He could be guilty but I struggle to see when- unless he is not going to work as OP thinks or it was during the night out with his brother. Or it's someone from work, I suppose.
Re the internet history and not being logged into Facebook being suspicious- possibly, but maybe not. On this basis, my partner could be suspicious of me. I log out of FB every time and periodically wipe history/remove cookies etc (sometimes weekly)- I have nothing to hide though. I also have my phone password protected and he doesn't know the password. Because I don't see that he needs to know it. Equally, doesn't know my computer password is. I am not cheating....
Also "keeping an eye on the condoms in the drawer" may not prove anything. He could get rid of them because he's been cheating and is trying to cover his tracks. Or he could know they are there (if there in his underwear drawer he may well be fully aware they are there but just hasn't thrown them out) and think "if DP is suspicious about condom in my pocket, what's she going to think if she finds these?" and get rid of them.
Ultimately, OP I would say you have to decide whether you trust him or not. If you're genuinely not sure, I'd say that as you know him better than posters on here and only you can decide whether you can live with uncertainty this has situation has thrown up. Though if you can't discuss your real thoughts and fears with your partner and/or simply don't trust him, I wonder if the relationship can work.