You can have an unlimited number of IPs with the same DNS on a single machine, you can even dedicate more than 1 IP to a single LAN card.
What you cannot have on a single rig are multiple Gateways. That confuses the PC as to where to send and where to receive from.
If you want to use or gain access to 2 or more different networks from one rig with multiple LAN connections then you need to Bridge all the connections, but, in doing so you are compromising the securities of all the networks connected.
A network Layout will certainly shed more light on the issue and what is it that you are trying to do will be a helpful hint.
Teaming is good but has it's own drawbacks and may not work all the time.
There is not point in having Multiple connections to the same Router. It's not going to increase the speed unless you Team the cards together.