Frankly I expect you'd be fine with a Pentium or Core i3, that's still what I'd consider "light desktop usage". A faster chip couldn't hurt, but you may not be able to tell a difference between a $70 Pentium G4560 and $350 i7 7700K with your uses.
An i3 7100 or even Pentium G4560 would cut your budget down a bit. The Pentium is the same as an i3 but with slightly lower clocks and no AVX instructions.
Frankly I expect you'd be fine with a Pentium or Core i3, that's still what I'd consider "light desktop usage". A faster chip couldn't hurt, but you may not be able to tell a difference between a $70 Pentium G4560 and $350 i7 7700K with your uses.