Can this man be trusted again?

My boyfriend and I have been dating for 7 months. I have a key to his apartment( as of 3 weeks ago) and was thinking about giving him a key to my house. I learned my boyfriend is a serious functional alcoholic. He even drinks in his work truck. I have agreed to continue the relationship because I love him and he promises he is going to seek help . In one of his drunk nights he mentioned that a female from his past wanted to pay him for sex and gave me her first and last name. I called the female( using his phone) She informed me that she was just over to my boyfriends apartment, because he picked her up to have sex last month and that he paid her as he has always done for the last 5 years, on and off. She even told me that his son ( who moved in 2 months ago) took her home because my boyfriend was too drunk. I called his son to ask if he had indeed taken her home. He confirmed. When I asked my boyfriend about the incident, he lied until I brought up certain information. He then confessed that "yes he had been drinking, he picked her up to give her a ride somewhere but brought her by his apartment to chill for a while. He said that sex might have been on his mind, they sat in the bedroom and talked( but could never tell me about what), but then he came to his senses( in his drunken state) and told his son to take her home, because he knew he had too much to drink". All of this occurred while I was working on my second job. Since then it has turned me into an insecure snoop. While investigating him, I learned that he was still texting, calling, and emailing one ex (from my house) who is married and lives with her husband in another. I learned that he was keeping nude pictures of another ex and was still talking to her. I also saw text messages in which he was asking his daughter to tell her female neighbor he said "hello" and that the neighbor was "his speed". Lastly, another ex, sent him a text message that said, "she needed some d". He told me he would cut all the women but 1 week later of feuding and no loving, the ex who left the "needing some d" text was at his apartment when I made a pop visit. He told them to leave, then I left, but the next morning I saw an open wrapper of "black ant" in the trash. To show me he was serious, he called all the women in front of me to end it. what do you think? Please be serious and not cruel with your responses.