1) As fas as, i understand that when i put a transaction I sign it with my private key and remote party verifies it with my public key. Thus with brute attacks theoretically it is possible to find my private key. As I think institutions like NSA may have enough resources for these purposes. So it is possible to lose my bitcoins to these large institutions?
They can't. Don't worry about it.
2) Bitcoin block database is growing as transactions done and blocks found. So this file will be getting bigger and bigger in time, as it holds all transactions from launch. When bitcoin gets a full traction, what will be size of those files. Will they be manageable? At the moment they total around 300mb.
There's some magic using merkle trees that allows the database to be slimmed by tossing unneeded transactions. Currently this isn't implemented in the client, but it's possible.
3) Hashed block structure seemed too simple. And with the nonce rule, it may seem the proof of work, because calculation takes time. With time factor it seems solid. I wonder, whether it is possible to use these calculated hashes as lookup table for another purposes? That they may be using our computer power to calculate this lookup table while we are distracted as this. I know this theory has no solid grounds that project leader could not expect this popularity two years before hand. I am just curious.
The hashes are derived from rather mundane data: block headers, transaction information, etc. I can't think of what mischievous purpose they could be used for.