The calliper pins should be greased with brake grease after each change. I've forgotten to do that once on an older car and had some issues. It could be the pins aren't sliding smoothly at low speeds?
The rotors should either be changed or resurfaced each time as well. The old pads wear the rotors oddly sometimes, so the new ones don't make perfect contact, leading to issues. You 'can' sort of get away with this if the brake rotors show very little wear. (As per vini_i's note in the comments too).
The brake shims should have been changed too, if you can see them they should be shiny and new. Just allows for the pads to move as freely as they should.
I think generally, you're going to have some play in the calliper, it shouldn't be excessive. But if you look at a diagram of the brakes the callipers are almost floating on the pins.
If you quickly touch the hub / calliper area after a drive and it's super hot, or start to see 'blueing' on the rotors. (or smell a burning smell from the brake area) I wouldn't drive the car except to a shop to get it dealt with. Or do it yourself. Changing brake pads and rotors can be a pretty easy job with a manual and/or youtube. If you have some basic tools. Then at least you know what was done, or not done.