Paying rent in cash with reciept and contract?

It is certainly unisual, but not illegal to pay rent (or ask for rent payment) in cash.
Almost certainly there is some dubious reason to it (as is the case with basically all cash transactions above a few hundred Euros in Germany), so be careful!

Don't worry too much about the cash request and it is usually not dubious at all. It is quite usual to buy/sell used cars from/to private owners with cash. That is much easier than using other means.

Legally possible but be aware that the landlord, in this case the main renter, needs to sign the forms but does NOT register for you.

nz7521137: You are unfortunately wrong! Used cars are often paid in cash precisely to avoid legal requirements for warranty and taxation - not at all in the interest of the buyer, but usually the seller is willing to offer a lower price that way (for exactly that reason!).