@Lettie,
Lettie, pack up his stuff that he left at your house. Put the friendship ring he gave you, in an unmarked envelope and put it in with the other stuff you have packed up. Overnight the package to his house, with no note.
He has accused you of messing around several times now and whatever his triggers are for acting like this every so often, as long as you are truly not messing around or flirting with anyone else, then his behavior is unjustified.
It is understandable - his feelings about his wife who passed away, his children, his friends, his family - but they are not acceptable excuses for his treatment of you .... pulling you close, pushing you away, over and over again.
If he has emotional issues and cannot decide if he wants to take charge of his own life and try to be happy, then he should not be in a relationship at all.
I know emotions are involved on your part and you are probably already in love with him, but he has to learn a lesson and he has to learn it hard and fast. You will NOT put up with this behavior.
Cut his access to you right now. Return everything you have of his to him immediately. Do not take his calls, answer his texts, read his emails, use the same chat-sites where he may be (including looking up his facebook page if he has one).
When an adequate amount of time has passed - a week or two - whatever feels punishment enough, pick up the phone on his next call and explain to him that if he wants to be in relationship with you - it is all or nothing. He will invite you to his house as a guest or he will not stay at yours at all. He will introduce you to all of his family and tell them you are his girlfriend and be open and honest with them, or he will not be allowed to see your children/family. He will begin couples counselling on the issues of accusing you of infidelity, or the relationship is over completely.
Unless you are willing to accept his behavior (and some partners do) for the long-term, then you have to set some ground-rules apparently. His behavior has been inconsistent over the last 9 months and he has you dangling like bait on a string. Enough! You are going to have to get tough here and if he is not willing to work with you, then you know the relationship may not be worth salvaging.