I would recommend that you dry out the babies bum after wiping them down but you don't have too.
However the baby will be less prone to rashes and be more comfortable if the babies bum is dry when putting a new diaper on them.
Yes you should wipe your baby down after even just a pee diaper.
Even though diapers such as the disposable diapers absorb the pee very good and hold the pee away from the skin there still can be some wetness on the babies skin which should be wiped clean.
It's best to always at least use one wipe to wipe down the babies skin to keep them clean of pee because as pee gets on their skin it turns into ammonia and begins smelling and then can lead to rash.
I've always wiped my babies and toddlers bottoms down even when it was just pee and sometimes when they pooped I would use the shower or give them a bath to clean the poop off them.
That is unless I was out away from home then I would need to use wipes to wipe the poop off and sometimes the poop would be so bad that some would sadly be left over until we got home and then I would shower them off or at least bathe them.
Keeping the baby or toddlers bottom clean and free of poop and pee helps prevent diaper rash which is painful.