Is it somehow possible to activate ip forwarding only for one specific network adapter? Do I need to activate it for all of them and then somehow controll it with firewall rules or is there another (maybe simpler) way to specify it?
Unfortunately no, it will have a similar effect as bridging two connections. Windows selects the both interfaces based on the default gateway setting. The connection which it forwards (Bridges) on should not have a default gateway. It will then route that connection of to the first available connection which holds a default gateway.
If you wish for more advanced ruling and routing you'll have to use Router and Remote Access Service (RRAS) which is available on the server editions. Note however that the Core editions does not incorporate the RRAS Role.