Bitcoin adresses seem not to be collision resistant. Please tell me I am wrong
The public key is derived from the private key. Generating a private key is pretty collision resistant as the space is huge. The fact that an address is in turn a hash of the public key doesn't change that.


  So how do you prove, that you are the owner of a bitcoin adress?


By having the private key: only then can you sign transactions.

Consent of the receiver is not required to send a transaction. If the receiver really doesn't want free money, he can chose to not spend it.