Other anons will have better advice, but I think it's perfectly reasonable to be uncomfortable that your bf texts his ex gf regularly. Sure, it's been months, but she did dump him.
If he brought it up and seemed open about it, ask if he will pull back a bit from their relationship/social media conversations. I think that if you want to stay friends with your ex, you have to expect you can't be that close anymore.
I'm not sure what to else to say about following her on twitter/insta because people can use those differently (like frequently post more personal things, or just nice pictures from weddings, etc.). Are you three in the same friend group or see each other irl regularly?
Anyways, hope your bf is understanding and you guys work it out!