Someone clearly needs to grow up and start taking things seriously if he’s in a relationship. Most or even all men look at other women, same as most women look at other men. As long as they’re both just looking, that’s fine.
But to add, people change, but very rarely, if he’s messed up before he’s likely to mess up again (which I’m assuming was the reason for the initial split - just a wild guess). Give him a chance - and if he breaks your trust, dump him.
P.S. as for checking ones texts/messages - if you have doubts, do it. And if you find things that aren’t supposed to be there - well maybe someone doesn’t really care about you. I am the one for trust and all that schizz, but better be safe than sorry - I’ve had one guy text his ex to confess he’s still in love with her, while he was with me; and another one sextexting and receiving pics of his female “best friend” pleasing herself through snapchat. Neither of which I’d have found out otherwise.
But if you look and it’s all clear - relax, but still be vigilant. Trust is great, but honesty is better - and in my opinion if you are not happy with someone you should either try to fix whatever it is or leave them, however some people want to have their cake and eat it, instead of breaking up with someone who may not be for them they decide to just entertain themselves with other people.
Or, depending on the age and the mindset, they are just not ready to be in a relationship.
All in all, everyone makes mistakes, but if what they do is basically their lifestyle - if you don’t like it, you don’t really need to put up with it.