I try to connect a smart switch with an integral ground wire. There is no ground wire connected to my old switch. Can I connect the integral ground wire to the small hole circled in red in my photo. This small hole is the only one that fits my screw. The holes on the back side are too big. enter image description here
On metal boxes, ground wires go to the box first.
Switches with a metal yoke (thing the screws go through) automagically pick up ground from the metal box, via the mounting screws. You do not need a ground wire. (Which means in a smart switch, you can't test with the switch pulled out, if it is one that is UL listed to use ground for its internal power).
On some boxes there is only one ground screw. However yours appears to have two ground screws per section, and each screw takes up to 2 wires (they don't need to be looped, just torqued to spec). So you have an embarrassment of riches in ground terminals. Try to have the 2 wires reasonably match each other, e.g. don't put a #12 Romex wire with a #18 lead off a switch.