They are randomly generated are linked to every address on the wallet (a wallet can have more than one private key (or address)
Only your master private key is linked to every address in the wallet. Individual private keys are only linked to a single address.
A private key (or address) is used to authenticate transactions from a wallet address, while a public key (or address) is similar to an account number of your bank.
The words key and address are not synonymous. There is no such thing as a private address, and public keys and public addresses are two different things.
They usually start with the number 5 (on the main net and 9 on the test net)
Only if they have been converted to Wallet Import Format (WIF). A private key is more generally though of as a 64 character hexadecimal string.
This is a set of words that are randomly generated and are as secure as the private keys.
Actually, the 256 bit number known as the seed number is randomly generated. The words are very specifically calculated in order to encode this number.
which should preferably be permanently offline.
It should absolutely be permanently offline. As soon as your seed phrase has touched an internet enabled device, you should consider it compromised.