Search Results: good way to touch a girl

I have been with my boyfriend for over 6 months now and our relationship is going good. He treats me way better than my ex did. He shows so much efforts and would be there if I need him. He has told me about his past relationships and casual flings. So anyway, I have snooped around his phone (checking his text messages, fb messages) when he's not in the room. I have found nothing that serious. Only chats about few girls with his boys but dont see any messages from other girls (and if he did get messages from girls, he would tell me about it first). I can snoop around his phone because he has told me his password to his phone.
And when im not in the room, I come back and I go on my phone and I saw that notifications and messages comes up on the screen and I didn't touch my phone for few hours. So that means someone has tried to get on my phone. I never told my boyfriend my phone password so thats the furthest someone can go on my phone without knowing my password. So i figured maybe he tried to snoop around on my phone.
Is this a problem in our relationship? Were we wrong to do this?

This girl is a good person, she gets good grades, mostly A's, sometimes B's. The girl is single.
Things he has done:
-Stares in her eyes only while teaching. Never looks away.
-Makes a joke and looks at her quickly to see a reaction.
-If she laughs, he smiles at her and nods his head multiple times.
-After giving work to the students, he just sits at his place staring at her, watching whom she talks with.
-Hasn't touched her in any kind of way.
-Has stayed with her in a school road trip.
-Always notices her first when he enters the class and smiles at her.
-Stares at her weirdly (more like a glare) when she talks with another teacher.
(The girl has saved him once from getting burned, she came in front of him before he could get burned, making herself getting burned)