When using a hardware wallet, the gui wallet on the computer or phone creates the transactions, but does not sign it. The transaction details get sent to the hardware wallet and it signs it and sends it back? Is this correct?
Does the hardware wallet ever pass the private key data back to the computer to sign?
No. The private keys never leave the hardware wallet. If they did, that would ruin the purpose of a hardware wallet.
Which hardware wallets works with which software?
Electrum supports all of the commonly used hardware wallets (Trezor, Ledger (all products IIRC), Digital BitBox, and KeepKey). Each hardware wallet also has a manufacturer provided wallet. These wallets are usually web wallets or browser extensions. Many other wallet software support some hardware wallets (usually Trezor, Ledger, and KeepKey).
Which hardware wallet is the most trusted out of them?
Trezor and Ledger are the two go-to hardware wallets that people generally trust.
Can you import existing paper wallets or basically just private keys into the hardware wallet? And which ones support this feature?
No, hardware wallets generally cannot handle imported private keys. This is because they actually only store one private key and then use BIP 32 to derive the private key that you actually want to use when you sign something. They don't have much onboard memory to hold more keys.