Do NOT give him the ring. It is yours. It has no meaning to it - just value.it has nothing whatsoever to do with your relationship with him.
I think it is important to marry if you want children together but that's me. I was married for 25 years & he left in a heartbeat - vows or no vows - it gives you no guarantee.
This guy is making you happy - why leave a good thing.
I don't believe in ultimatums but I also don't plan my life on superstitions and signs.
Only you know whether you want to stay or go but ou cannot force him to marry ou & if you push - he will resent it.