Can I use Multibit private key for paper wallet? I plan to export the private keys from Multibit, keep those and delete everything else.

Now I read about Multibit having some kind of hidden mechanism for spare adresses within a wallet. Would my approach still work or do I risk losing the BTC in the wallet?
Yes, writing your MultiBit (non-HD) private keys on paper should work.

You definitely need to write down the private keys corresponding to public addresses that have a positive balance.

You also need to write down the private keys for public addresses that are present in your MultiBit wallet but have not been used yet. This is because they are considered to be on standby, and may be used in the future for receiving change or incoming payments.

As long as you carefully back up all your private keys and don't generate any new ones, you can always restore your wallet just from the private keys alone (with no other metadata, etc.).

Be warned that whenever you create a new wallet in MultiBit, it will generate one random key for you by default. So even if you import your old keys into the wallet, you will still have that new fresh key which needs to be backed up, otherwise you risk losing coins.