Let's say I save the private seed of my Exodus wallet, how can I be sure that in 20 years from now, Exodus wallet will still be a working product, that I can download from a new computer perhaps and import my seed into?
You can't.
What you can be sure of is that there will be some wallet software you can import your seed phrase in to. Most wallets which generate seed phrases,* Exodus included, generate them following the
BIP39 process, which is well known and well documented. You can take a BIP39 seed phrase from any wallet and import it in to any other BIP39 compatible wallet and recover the same wallet with the same private keys. Even if every piece of wallet software on the planet no longer existed, and for some reason you could not access an old version of any wallet software, then the process to turn the seed phrase in to private keys is still well document and it would be fairly easy for you to recreate your private keys if needed.
The private key of my Bitcoin wallet is different ofcourse, but I do wonder, will this private key allow me to import the address into other wallet softwares? Or do I have to import it on Exodus?
You can import a private key in to any wallet which supports importing individual private keys. Private keys can be used to generate different types of addresses though (most commonly legacy (P2PKH), nested segwit (P2SH-P2WPKH), and native segwit (P2WPKH)), so you'll need to make sure the wallet you are importing to will generate the right address for you.
One thing that also slightly confused me was the fact that when I request my private key from the Exodus wallet, it shows me multiple private keys and public wallets. Two public wallets starting with 'bc1q', and one wallet starting with '157okQ'. Which one would I have to save? When I make a request to receive funds in the wallet it shows my deposit address as one of the bc1q addresses. Will this be the address of which the associated private key is practically my money?
The addresses which begin with "1" are legacy addresses. The addresses which begin with "bc1" are native segwit addresses. It is preferable to use the segwit addresses as they result in smaller transactions and therefore lower fees. In terms of saving your private keys, I wouldn't at all - I would just write down your seed phrase and be comfortable in the knowledge that the seed phrase can recover all those private keys in the future if you need it to.
*Electrum is the most significant outlier here, which uses it's own seed phrase and derivation method. But again, this is widely known and publicized, so even if Electrum no longer exists in 20 years, it would be easy to convert the seed phrase back in to your private keys.