Shy guy's opinions, is he into me?

this really shy boy that i like has been opening up to me more and more lately, i sit across from him at lunch and before he was very quiet towards me and wouldnt really talk much, and a friend of mine told him i thought he was cute and he replied with "no she doesnt!" and a smile, after that he kind of tried to avoid sitting near me at the lunch table but i talked to him and i think hes feeling more comfortable with me, do you think he was acting like that because he didnt like me or because he was just nervous because he knew i thought he was cute? the past couple days have been a lot better... we were just stealing eachothers things off the table like he stole my waterbottle and i stole his planner and he let me keep it overnight and i drew a bunch of stuff on it and he took mine today and wrote "this is for writing in my planner" and scribbled and was laughing and gave it back, we also had a small conversation... I said "whats up ____" and he said "whats up ____" and i said "nothing much" and he said "oh... so are you going to the game tomorrow" and i said "oh i dont know yet" and he said "oh... im not going". do you think he asked because he may have wanted to hang out with me? also im planning on asking him to come hang out with me while i babysit my cousins tomorrow night... do you think he might like me... if he isnt busy and can come hang out with me tomorrow maybe he will get a little more comfortable hanging out with me out of school? any suggestions or comments on this? Do you think I should ask him to come hang out with me while i babysit or no?