I know what it's like to have parent's that are rigid when it comes to certian things, and I imagine it would be even harder living in a Catholic family. Are you the firstborn, is that why they punish you the hardest? Most times that is the case, and being the firstborn also has the virtue of the added pressure to "take up the family business." A lot of times I've felt that my dad was unfair to me, and it's tough to voice your opinion without getting the big talk, but some things you just can't talk to your parents about initially because of how they'll react. Of course this depends on the situiation. I personally think you should seek out a leader in the Catholic church and find their take on it. Some one you trust who thinks the way you do.
To tell the truth, most people fall away from their faith during thier teenage years for a number of reasons. Most being unsure, life changing occurances, and other reasons. I myself went through a period where I wasn't sure if God was real to me. I was very depressed and it seemed like the whole world was against me. I couldn't see past all the bad, all the ugly, and my life felt devoid of warmth and love. My depression was linked to events earlier in my life, involving a divorce, among other things, but I see it the same as what all teens go through some time, and it was just as hard as what anyone goes through.
I eventually got through it, though even now I still sometimes struggle with it. It doesn't help that my dad is against using drugs and therapy. He thinks I should do it on my own... But that's not the story at hand.
I would like to encourage you in your faith, and it seems to me you want to believe, so I know the desire is there. It may feel like God is neglecting you at times, but he's still there, still loving you. You might feel barren of emotion, or that the whole world is against you, but do not be discouraged. There are people who love you, who can help you through this. You can't analyze God. My dad kept trying to his whole life and kept running into roadblocks. It can't be done. I think I read that you wanted to see his existence proven, but if we know God exists through pure factual evidence and it's concrete that he exists (i.e. if God was testable like the outcome of scientific hypotheses) then what would having faith in him mean? It wouldn't mean anything. It would be pointless to believe because it was already proven by facts and evidence that you could clearly see and believe. But it's not that easy. You have to "walk by faith, not by sight." Do not lose hope or be discouraged, for the Kingdom of Heaven is yours if you only ask.
EDIT: Don't flame this, I'm just expressing my views via freedom of speech. Also, if you need something elucidated, feel free to have me explain.