My take is that you should just ignore their request; you're obviously required to return equipment and confidential items when leaving any employ, so you don't

My take is that you should just ignore their request; you're obviously required to return equipment and confidential items when leaving any employ, so you don't need to sign a contract for that - it's dumb for them to ask. The other requests are stupid and pointless - so don't get into it with them. HR departments are almost always useless SOBs; hell, if they were any good they would figured out how to stop you from leaving in the first place...