I'm super jealous of my boyfriend's ex because they still talk all the time.?

Okay, so first things first: I found out all this information from snooping, which I know is awful and basically I got what I deserved. In your answer feel free to address and analyze this, but also give your thoughts on the next part.
So through snooping I found out that my boyfriend still texts with his ex a lot. Their relationship ended a long time ago, about 2 years, but they were dating for 3 years (we've only been dating 5 months). All the texts are innocent and just jokes and stuff, but it still makes me a little uncomfortable. Also he has kept gifts she gave him when they were going out, like love notes and stuff, but he does keep them out of site. Now he's only ever mentioned her in incredibly brief passing and never brings her up, he hasn't even mentioned her name ever to me (but I figured it out through the aforementioned snooping), and I don't know whether that's a good sign or a bad sign. Also, she lives in a different city so they never see each other.
The worst part is, I can't really confront him about it without admitting to my snooping, which is clearly a violation of trust and maybe a relationship ender? When I have brought up ex's in general he has always stated that he's happy to be with me and I'm the one he wants to make it work with.
Basically I'm just wondering what you guys think of this situation. Do you think he still has feelings for her, or is it really just a platonic friendship? Do you think I should bring it up or will that only make it work?