Is it possible to move my wallet.dat directly to a hardware wallet such as Tresor? I am using Bitcoin Core and want to move my wallet off my PC to more secure storage.
Thats completely against the purpose of a hardware wallet.
Once your private keys were on an device connected to the internet (your current pc), those should be considered as possibly compromised.
The only way to guarantee a safe storage is to let the private keys be generated by your hardware wallet (which will never leave the device) and send your funds over.
Also, it is possible to access my wallet in Bitcoin Core without downloading the whole blockchain. My client crashed recently with a message that said "Error: a fatal internal error occurred, see debug for details" .
Not directly.
You can export the private keys / the master private key (starting with xprv..) and import them into another wallet (e.g. electrum:
https://electrum.org/#home).
To export your master private key in core. Go to
'Help' -> 'Debug Window' -> 'Console'.
Then type in
walletpassphrase YOUR_PASSWORD to unlock your wallet and
dumpwallet FILE_NAME to dump your allet into a file.
Afterwards you can open this file with an editor and serach for 'xprv'. This master private key can be then imported into any BIP44-compatible wallet.