If you have the basic requirements for a relationship then you shouldn't buy anything or do anything permanent. You do not have fidelity trust or communication. You are not in a position to consider property. Sharing a living space isn't enough. No one should be "starting" arguments to get to do things they want to do. That is a sign that the communication is not what it should be. I would also advise that you avoid talking to kids about your relationship versus their mother's. You have to decide if this relationship is strong enough to have the kind of bond that does not involve these kinds of problems. Lying and cheating alone is not a healthy relationship. If you are a revolving door you have to make changes that puts an end to that. You shouldn't' tolerate anyone treating you this way. You can always end the cycle. If you want to stay then consider a couples counselor or developing communication. He has to decide what he wants too. You can't be somebody that he returns to when he wants to.
Consider counseling. Find a copy of Mars and Venus Together Forever. Protect yourself from any problems down the line by not entering into anything permanent. Sit him down and talk to him about your concerns. Consider couples counseling online if he is resistant to going to see a professional.
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