If you see some of the pads are a very dull grey from leaking batteries, try scraping them clean first before adding new solder, otherwise the solder may not flow. I will always apply fresh solder before desoldering because it cleans the contact allowing better heat transfer and thus less time to desolder the pin.

I think the pads lift when desoldering in part due to the adhesive holding the trace/pad to the board loosening from the heat, and in part due to physical pressure and motion.

It's best to suck the solder out first, applying as little physical pressure and motion as possible, then let the board cool so that the adhesive is strong again.

To break any remaining solder that might he holding a component pin, I like to use a needle nose pliers to hold the end of the pin (from the backside) and wiggle in a circular motion (don't pull on the pin as you may yank out a feed through that makes contact from the front to the back of the board). You can usually tell when the pin is moving freely in the hole. If it doesn't move freely that means there is still too much solder holding the pin (often happens on pins connected to a ground plane). If that happens, apply fresh solder and try again. If you still can't remove enough of the solder, then cut the pin and remove it individually.

The component usually just falls off the board when all pins are free.

Looks nice Chris!