You have to make sure that the clips on the back of the pad are seated correctly in the groove of the piston or like you say the pad won't be sitting correctly, this in turn twists the caliper slightly when you apply the brake so the pads aren't contacting the disc fully.
While you have the caliper off it's a good idea to remove the rubber bushes that the guide pins go through from the caliper and clean any corrosion from the caliper where the bushes sit. Water gets between the bush and the caliper and causes corrosion, this builds up and eventually causes the rubber bush to nip onto the guide pin and then the caliper doesn't slide freely like it should.
A coating of silicone grease on the rubber bush where it sits in the caliper will help stop any corrosion.
If the handbrake levers of the caliper are stiff it's probably time to think about either a replacement / reconditioned caliper or striping & overhauling the one you have.
You can sometimes free off the lever mechanism with some WD 40 or similar but it's short lived in my experience.
I overhauled the calipers on my car a while back as the handbrake had reached the point of hardly working, despite having new cables, discs & pads and pulling the handbrake on with more force than the Incredible Hulk could manage it still wouldn't hold the car on a hill.
After the overhaul the handbrake was (still is) great, it holds the car fine on a hill and you don't need super human strength to apply it.
The caliper overhaul thread is HERE
if you want to take a look.
sell all of the seals in a kit for the rear calipers. (They also sell on