Yes, they will work for your application. Yes, a TS2 is a temperature switch, thus the "TS." Brutus 10 clone builders usually use the TS-13010, but your TS2 will work almost the exact same way: the specs are barely different, and it has a different wiring scheme.
A PID can be used for a wide range of industrial apps (much more than a TS), but you don't need a PID. A PID will work just as well for your app, but a TS is all you really need for your direct-fired RIMS app.
BTW, nice website!