Is he really over him ?

I have been with my boyfriend for a little over three months now. The relationship has taken a serious turn when we moved in together after three months. Now seeing how this is not normal for me to let these things happen so soon, I would be hesitant but this relationship has been the most intense and one that I actually care about more then the rest. I understand that communication between two of his ex’s is still there including his most recent, I do feel he loves me entirely but last night a belated Christmas gift came with a few things from his ex, to a house we both live in….. Which means he’s giving out our address to them , allowing the opportunity for me to feel hurt. Now i feel perhaps I am being overly considerate trying to make sure he doesnt experience and of the risidual shit from my past, but its when i asked about the gift all the lies started to happen. I asked who it was from , got a lie, after opening the gift infront of me he proceeded to wear the hat that was sent around the house in front of me. When we went out for a smoke I asked if that was the hat he sent , he said no , another lie, and then all this led to the question ; Does your ex even know about me ?. The responses was yes, survey says….LIE…..... I feel very defeated and almost as if i am battling with someone who doesn’t even want me to win….. He says he wanted to spare my feelings , but lies hurt way more then honesty ever could cause now I am questioning things….... We live together , hes not from here I am under alot of pressure to have to figure out things in short amounts of time and am feeling very scared with regards to what the possible outcome could be of this…...any advice ?