I created a vanity public key (wallet address) and I got the corresponding private key too. Is it possible to somehow "open" that wallet and just used it as a normal bitcoin wallet? No sweeping, no transferring to other account - just send some of the money from that account to some other.
A bitcoin address is not the same as a wallet. If you have the private key for the address, you can import the key into either Bitcoin-Qt or a lightweight wallet such as MultiBit. You will then be able to send and receive coins using this address.
To import a private key into Bitcoin-Qt, read this
To import a private key into MultiBit, read this