That he would be meeting her family before she meets you doesn't seem right. That says to me, that he isn't planning on staying married. I would say your marriage is in more trouble than you or possibly even both of you realize or will admit to. Take a real hard look at the things he has told the girlfriend, that appear to be lies, and look at it as a BIG RED SIGN that indicates his state of mind.
When my marriage was falling apart, my husband would tell me where he was going, but I didn't always believe him. I resented that he was gone all the time and there was past cheating issues, so my mind would invent all sorts of things that he was doing instead of what he said he was doing, especially if he was late coming home. This was not rational, nor was it reality, but it was a warning sign of big troubles in our marriage.
Don't ignore these things! It looks like you guys have a lot of work ahead of you to repair things between you.