Should i stay in my relationship?

ok sorry folks this will be a bit of an essay!! i am with my boyfriend a year now. at the start things were brilliant. an ex of his was still friends wit him and i had no problem with it. when we got very serious she started texting him and kept ignoring me. so he kept texting her problem one! problem 2 i caught him texting another ex girlfriend but i only seen the messages because i was texting a friend off his phone,seen her name and then looked at them..i got angry with him because he never said he was texting her to me [he was drunk while texting her by the way] he didnt see a problem with this,he thought it was normal..i went to bed and he kept texting her..the next day i checked his phone and he deleted all his messages. my boyfriend is a bad drunk and another time we were out with friends he got really drunk and started going mad for no reason..he then pushed me very hard when i told him to calm down. when we got home that night i was crying and asked him why he pushed me..he said very smartly you should just forget about that! he didnt apologizee at all. other then these problems he is never very loving towards me unless i start hugging or kissing him first. i have never done anythingn in the whole year to make him angry or upset him. i wonder should i stay with him or go but i love him dearly. i just think maybe there was too many negative things happening after all the first year is suppose to be the ''honey moon period''. need some advise so any comments would be greatly appreciated!