Yes they exist, i3, i5, i7, a10, a6, a4, and a few other older generation parts not to mention laptop cpus. I think you are thinking PSUs to which I point you to pico psus that use a twelve volt input and are tiny.
If you want to use the integrated graphics for light gaming, the A6 is probably your best bet. None of intel's cpus under $100 even have the HD4000 which is still worse than the A6.
Obviously the A10 is better but by how much. This is a very subjective question but what would feel faster, an A10 with a HDD and 8gb of 1600 ram or an A6 with a SSD and 8gb of 1866 ram.