There are some hazy lines here, for example telling him to message her and show you that hes blocked her is a little controlling, if my GF said that to me I would take a step back and review our relationship. You either trust him or you don't, if you don't - get past it or move on, if you do - don't tell him what to say to other people, a little creepy.
On the other hand he sounds like a bit of a dick that hes texting other girls behind your back, I wouldn't trust him personally, as it shows you don't, but then you have to wonder if you can get used to this? Guys don't get 'better', you aren't going to be able to 'Change' him, and if he's unhappy enough with the relationship to be seeking things elsewhere, then he obviously isn't committed as much to you as you are to him.
Theres tension as you are being controlling because of his actions, seems like a rocky relationship that isn't going to change much anytime soon.
My advice: Get out, move on, think its best for both of you to find happiness with other people. Irrelevant whether he is nice sometimes, you need to stop focusing on the good things as they obviously aren't good enough for you to be happy with the relationship.
Best of luck