Since the airgapped wallet does not know anything about which inputs to be used, then you will also need to copy across individual inputs and then manually craft your transaction since I don't know of any airgapped wallet which allows you to import individual inputs.
Indeed. You cannot import inputs in known wallet software. You can only sign unsigned transactions.
As I said above, although Sparrow lets you chose a different SIGHASH to use, it doesn't actually show you which SIGHASHes were used once the transaction has been created.
Sparrow doesn't show you which SIGHASHes are used in a signed transaction, but it does show that information if you import an unsigned transaction. So, a solution would be to install Sparrow on your airgapped device, and before you sign a transaction, you can open it up with Sparrow to check.
Although, I do agree that simply displaying the SIGHASH in the preview is fairly simple and informative.