I'd recommend an Ivy Bridge or Sandy Bridge E3 Xeon, such as the E3 1245. These CPUs are more or less lower clocked i7 chips, and have excellent performance. They will, on average, have a performance between a stock i5 and i7. As shown in these benchmarks, this means that at 1080p, you will easily lock 60 FPS for the latter two games. It is very likely you will be able to achieve a minimum of 60 FPS for the near future (ignoring GPU), except on very CPU heavy games like Assasin's Creed.
As to which Xeon to pick specifically, obviously, the bigger the number, the better. Note that the 1220 and 1225 series do not feature hyperthreading, and you may as well buy the equivalent i5. If you can find an equivalent i7 at under your budget, that would also be an option. Ivy Bridge (v2 or 3xxx) CPUs are more or less equivalent to a Sandy Bridge CPU clocked ~ 100 - 200 MHz faster.
These CPUs require a LGA 1155 motherboard. Check the manufacturer's webpage for compatibility if purchasing a Ivy Bridge CPU.
Newer platform, similar performance: The Ryzen 3 1200 is just under your budget for $99.99 at Amazon.com. At stock, it's about the same performance as a higher end IB Xeon but you do get access to some modern bells and whistles on the newer platform. It has an average overclock of 3.9 GHz, which may give a decent boost to performance. On the other hand, the newer platform has higher costs associated with the RAM and others. The Ryzen CPUs use a AM4 socket.
The new Coffee Lake i3s are just out of your price range, but may also be worth considering. The old AMD FX series on the other hand, while a decent amount cheaper, $75 for a "6 core" and $90 for 8 – and have attractive looking clock speeds – are definitely to be avoided, as they have significantly lower performance due to low IPC.