I understand that flaps are devices used to increase the amount of lift and drag. When landing with a tailwind your groundspeed will be higher. Therefore, flaps would help reduce this groundspeed, logically allowing for a lower touchdown speed.
Should flaps always be applied with a tailwind? Are there reasons not to use flaps when landing with a tailwind?
Forget about tailwind/headwind for flap use. Flaps allow landing with a lower airspeed. And you always want a lower airspeed. The wind itself is irrelevant except that your groundspeed will be higher or lower depending on where the wind is, and which you always want to be slow as possible. On light aircraft you want to avoid tailwind components over 5kt. Jets are usually certified, takeoff and landing performance wise, with a max tailwind component of 10-15 kt. But in any case you are still using normal landing flap to have the minimum airspeed, giving the minimum groundspeed for a particular condition.
The time when you may want to land with less than landing flap is when the winds are strong and very gusty (and you will be landing into wind if you aren't crazy) and you want to have a bit more energy with a bit less drag (so the aircraft conserves more energy with its inertia) during the landing flare to cope with sudden small airspeed variations from gusts, which can drop you on the runway hard before you can react if you have the flaps all the way out.
On a windy day there is a typically a wind gradient, meaning that the wind speed increases with altitude. On normal into-the-wind landings, the wind gradient tends to exacerbate any accidental "ballooning" (meaning that if aircraft accidentally rises during the landing flare, it tends to keep rising, till it eventually ends up in a very low-airspeed condition.) Especially in gusty conditions. Flaps tend to aggravate this. When landing downwind, the wind gradient tends to reduce any accidental "ballooning" tendencies during the landing flare. On the other hand, downwind landings involve high groundspeeds which can tend to exacerbate steering problems, especially in tailwheel aircraft. Flaps reduce the airspeed, and therefore the groundspeed, at touchdown.
As a practical matter, when playing around with multiple upwind and downwind touch-and-goes back-to-back in windy conditions with a radio-controlled model airplane (most pilots would not be interested in doing such a test with a "real" plane!), in a situation where runway length was not a limiting factor and the criteria of interest were a smooth gentle contact with the ground followed by a highly controllable run on the wheels before liftoff, I've found that the best results were generally achieved by using flaps when landing downwind and not using flaps when landing upwind.
