Does my light switch have a ground?
Your light switch is not grounded and neither is the plastic box. You do have a ground wire in the box. It's the bare copper wires wire nutted together. To ground a switch, you'd need to get a piece of 14 AWG wire and connect it to the group of bare copper wires and then to the switch.
Most smart switches need a neutral, which isn't the same as a ground. The neutrals would be a group of white wires connected by a wire nut. Scrape some of the paint off the wires to make sure of the colors. Make sure the power's off before doing any work.