note: I might be wrong
The EVM is actually the core of Ethereum. All contracts are written in EVM bytecode. This doesn't mean that Ethereum includes an actual virtual machine in the sense of Hyper-V, Vmware or Xen. It contains something[*] that can execute EVM bytecode (that something can either be interpreter, jit or aot). So, VM in this case is just a ...
So, this smart contract, with its setup today, would it be possible to make it use the RSK network instead of Ethereum?
RSK Bridge can also be helpful for users to swap tokens: https://developers.rsk.co/tools/tokenbridge/
If yes, what could be the pros/cons (eg. cheaper/faster ...
They are completely separate networks, with a different genesis block.
Geth and parity support running your own private networks that can utilize the features of ethereum (and the ecosystem of tools for it), without having to interact with the public network.