How long have you been with your dp and do you live together?
In what way can it be said that he cheated on you? By unexpectedly bumping into this particular ex and giving her a kiss hello/goodbye?
If, as she claims, he did kiss her it's unlikely to have been any more than the type of token kiss that's commonplace when, whether planned or unplanned, those who know each other meet up in the street.
Or do your imaginings have it that their eyes locked across a crowded road and they launched themselves at each other whereupon she fell on his dick they engaged in a bodice ripping snogfest while passers-by gazed on in envy awe?
She's told you that when she asked for his number he told her "he was sorry but he couldn't as he had a girlfriend". Doesn't that tell you this woman has made contact with you out of spite because he made it clear he wasn't interested in her advances? It appears that you've had this confirmed by the friend who told you she was a troublemaker.
I'm not surprised he's hurt that you've chosen to pay attention to this woman. If I had a partner who put the word of one of my exes above mine I'd walk because, as you rightly say, a relationship without trust is hell and I wouldn't want be to around someone who didn't trust me.
Fwiw, I'd also be mightily peeved if they made a mountain out of a molehill by discussing something of so little consequence with their family and friends as I'd fear that they may be looking at me as if I'd got horns coming out of my head consider me to be untrustworthy.
Please seek professional help for your trust issues otherwise you could fall into the trap of spoiling good relationships because you've had a few bad ones.