THE EX FACTOR

OMG...I have a boyfriend who I love so much. However, he has an ex-girlfriend that he considers a "Good friend." I feel its nothing wrong with having a friend, but when I find text messages in his phone that he send to her saying how he loves her, and also text message from her calling him pet names such as "DAD" that's when a problem is set with me. Now I told my boyfriend I am not at all happy with that, and I don't trust what's going on, but he said he talk to the ex-girlfriend/good friend about the pet names, and it won't happen again. Then all of a sudden an emergency situation happened and the ex-girl was in the hospital due to medical issues, and she asked my boyfriend to house-sit for her. My boyfriend had no problem helping her and began to go to the ex house everyday to make sure the house was ok and the cat was fed. He even took me one time, and I was just watching to see how did he act around the house, and he knew the house all too well. He even went into the bedroom and was looking for some glasses, but that's when I knew he been in the bedroom more than he should've been. After that, I told him again how it bothers me that it seems as if he cares for this "EX" too much, but all he did was listen and tell me that he is sorry but he will not cut off his friend for me because they have more than 10yrs history and he's not that type of person to do so. In result to that, he even went to go visit the girl in the hospital after that. I was upset and again, I voiced how I feel. But in my heart I feel that I am his woman and the ex is a past thing. Its cool to be friends but it just seems like too much. I try to work with it but, every now and then I will get fed up because I don't like that they communicate so much. Also, just make note that his mail still goes to the ex-girlfriend house and I feel like he uses that as an excuse to see her and keep her content. I have my own place but when I asked him why he don't have his mail go to my place he said because he don't no how long ima be living there. I believe they still do undercover things because I remember he told me the ex girl told him he can come back home at anytime. Now what is wrong with that picture?
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There's a few things wrong with this picture. It is not at all uncommon for a couple to break up and remain friends, in fact, if the couple remains friends , that means that it was probably a mutual break up.  If it were a break up that occured as the result of a fight or disagreement where one person was angry and called it off, it would be more likely that they would patch it up and reconcile. however when there's a mutual break up it's usually because both parties have decided that it's not working, and they've made their peace with it.  It's important that you let him know your concerns about the nature of their relationship, without giving ultimatums, people generally don't respond well to them especially not men.  Try setting boundaries instead, and letting him know that certain things like her calling him "dad" has to stop, because you feel disrespected.  The issues about the house sitting ( "him knowing the house all too well"),  and the mail, are not the real problem here.  So don't argue back and forth about these "issues", get to the root of the "problem", which is that you think he's cheating. period.  If he was willing to take you to the house with him, and agreed to talk to her about the "pet names", then it sounds as if he's trying to address your concerns, as long as he continues to do that, and unless you find concrete evidence that he is in fact being unfaithful, trust your man...but still keep your eyes open.