How to determine what's mine? I for example own 10 BTC does that mean, in some Blocks in the valid part of the blockchain is written, that I received at some point in time a certain amount of BTC? How do I prove that the Transactions in this block belong to me (without loosing ownership).

How hard is it to find all transactions in a chain which belong to me. Some how this should be possible in $\mathcal{O}(l\cdot m\cdot log(n))$ because of the merkle tree in each block for m blocks and l transactions representing my complete ownership (3 BTC + 4 BTC + 3 BTC = 10 BTC)

Could everybody check my amount of bitcoin if he knows my alias or whatever is used to mark my transactions as mine. 

Could I just spend someone elses coins?

Thank you very much in advance.

PS.:
My Plain written math expression was displayed wrong so I edited as tex code but it still does not work. Anyone who can read tex will still be able to read this formula (probably better than before)
If sb knows how to fix it, plz provide a solution.
I for example own 10 BTC does that mean, in some Blocks in the valid part of the blockchain is written, that I received at some point in time a certain amount of BTC? How do I prove that the Transactions in this block belong to me (without loosing ownership).


Essentially, yes. Owning BTC simply means that one or more transaction outputs somewhere in the chain exist, that are spendable by you. In a normal case, this means that you have control of the private keys required to spend these outputs.

To prove you own them, you can sign a non-transaction message using your private key. This signature can be verified against your public key, and the public key can be verified against an address, thus proving that you own the outputs associated with that address.


  How hard is it to find all transactions in a chain which belong to me


If you do not know the transaction IDs, the merkle tree will not help. The merkle tree is only useful if you want proof that a known txid is in some block. If you only know the addresses that your outputs are at, but not the txids, you must scan the entire chain and match the output scripts against your addresses. This will have a worst case of O(n).


  Could everybody check my amount of bitcoin if he knows my alias or whatever is used to mark my transactions as mine.


If they know which addresses belong to you, yes.


  Could I just spend someone elses coins?


If you are able to satisfy the output script requirements, then yes. For regular transactions, this would imply that you have been able to compromise the keys owned by someone else.