The customer paid on their own without me sending them a payment request. I didn’t see any type of notification so I didn’t realize they had.

You should have received a booking request e-mail from Houfy as well as a notification on your phone. Both direct you straight to the messaging system at Houfy where you can accept or decline or discuss.

My question is they paid for the three nights but they’ve not paid for the damage deposit. So how do I get them to pay that?

Houfy does NOT collect damage deposits. (Due to the many different options owners have for this - collecting/refunding timeframes.

And is the reservation automatically confirmed since they paid? Once you, the owner accepts the reservation request, yes. Your calendar gets blocked. Both you and the guest will receive an automated e-mail of the confirmed reservation.

Please check your "spam" folder and mark Houfy as NOT SPAM!