door handle button sticks

whoa!! slow down-- you might need that clip, might not--- first off --door panel-- is held in place by strong clips that usually tear the backer board & proper tool & time they snap right out.
slide tool so said clip is in between the forks & pry-- should pop out-- screws are self-explanatory.

as for the push -button-- your door has seen a lot of use-- has it ever been cleaned/lubed?

try spraying it down with a ton of wd-40(won't harm the paint)
use some carb cleaner at first with the nozzle attached & spray inside the latch mechanism.
see if this improves the latch movement.
take your time & search on this site-- you can do it -- ,Lawrence^