Are private keys able to be transported between different hot wallet softwares? The answer is yes if you have 12 mnemonic or private key from one software you can easliy move to another software in you case be able to transport from the exodus to any hot wallet out there
Not every wallet though but most modern wallet , exchange wallets won't do that you can only follow thier protocols, it's not your keys and there's no way to import using keys ... HD wallets should allow the import of private key, but a private key should be limited to a single address I think, so having multiple address will require multiple private keys for each address, the 12 mnemonic seed will import the whole address if all the address were generated using the seed,
Exodus on the other hand, will give seed but the private key can be derived using tools I think(OP should just forget private key he seem to be confusing it with mnemonic seed in which they are different) .
Importing the seed from Exodus to another wallet software is just what he need like you mentioned...