channel-id is important for new subscribers to read my stream. How can I securely transmit them over the network? Shouldn´t be the key exchange algorithm also quantum proof?
You can use a Key-Sharing technique like a Group Diffie-Hellman key agreement or you can mask (encrypt) the sideKey and channel-id using the Public keys of the receivers. As i know, there's a public-key cryptosystem called NTRU that is said it has resistance to quantum-computer-based attacks. So i think this is maybe the solution to this problem.
How can I securely transmit the keys?
To get the key from one actor to another, you have to establish a connection between the two, where no third party can interfere with the messages sent along this connection. Cryptographers call this a "secure channel". There are various techniques used to achieve this secure channel, though the well established ones (RSA, ...) are not quantum proof.
Shouldn't be the key exchange algorithm also quantum proof?
Depending on your use case, you could use a quantum-nonresistant algorithm until quantum computers exist and upgrade after or immediately use Post-quantum algorithms.
In the future, we might also be able to use something like Quantum key distribution for key exchange.