Do either one of you ever believe what the other says? He says you think he is cheating, you tell him you don't think so. He tells you he didn't spend the night at her place, you call him a liar. He tells you the key that you are concerned about is not to her house, you call him a liar. You accuse him of talking to her all the time, he tells you that he does not. (He talks to her weekly, per your post, and you believe that is all the time? Seriously?) You get upset (rightly so) about a pet name in his phone and he does finally take it out then you are upset that he doesn't put your name in his phone at all? Then you accuse him of not doing so simply because you want him to put it there? (Are you sure that is really worth fighting over?) You get mad over one (ONE??) saturday night that he does not spend with you?
And as far as calling someone "beautiful" or "sweet girl" or any such thing, let me just say that I've been married for 34+ years and I still address women I am close to by calling out (or texting) "hey there beautiful." Thank God my wife is a bit less picky about this than you are.
Oh and I don’t know this for certain but I believe she doesn’t even know about me
You are correct, you do not know that for certain.
Look, you have some valid points, IF he spends the night at her place, IF he has a key to her house, IF he talks to her all the time (weekly is not all the time.) Maybe he just wasn't ready for a new relationship. Or maybe you are being so jealous that you figure he must be lying to you when he answers your concerns.
Either way, the bottom line is that ehBeth is correct. The two of you just are not going to be good for each other.
We are now ending our relationship because of this.
I think this is a good idea. Good luck.