Can a crypto belong to only one network?
I do not think a coin canbe operated on more than one network.
Where can you check network affinity?
A Google search of the cryptocurrency should reveal the network it is built on. You can use the suggestions above as well.
Must a crypto have different contract addresses in different networks?
I don't understand this question.
How can you check which crypto is the native token in a given network?
It commonly follows the name of the network, such as ERC-20; the 'E' stands for ethereum.