Well, stabilized lenses work just the same as non-stabilized lenses on tripods and monopods - if you turn the stabilization off (which you can do on all IS lenses). So that's not something you need to worry about. A few of the most expensive long telephoto lenses with IS actually have a system which turns the IS off if the lens is very stable (i.e. it knows it's on a tripod!).
The reason that IS and priods don't always mix is that the IS system is always trying to stabilize. If the lens is perfectly stable, the IS system may cause the image to slowly dirft, which can result in unsharp images if the exposure time is long (as it can be on a tripod)
The advantage of a stabilized lens is that you don't HAVE to carry a tripod or monopod, not that they are better than lenses used with a tripod of monopod. The most stable possible support is given by a study tripod, but a sturdy tripod isn't always convenient. They are heavy, you can't use them in crowed places and they're something else you have to carry. Monopods are a little more convenient, but don't give nearly as much stability as a tripod.
You can certainly photograph moving subjects with IS lenses. Most of them have two IS modes. One stabilized in both the horizontal and vertical directions, but in the second mode the lens can tell if it is being panned to trach a moving subject and it can turn off IS on one axis. For example if you are shooting a passing car by panning in a horizontal direction, the lens will turn off stabilization in the horizontal plane, allowing you to floow the car, but it will keep stabilization in the vertical plane to compensate for unwanted up and down motion of the camera.
So basically, the only reason NOT to buy a stiabilzed lens (if you have a choice) is cost. That only applies to two lenses because only two lenses are offered in IS and non-IS versions and they are the 70-200/2.8L and 70-200/4L.
Switching from IS to non IS does not in any way "give you many more stops of light" unless you buy a different non-IS lens, and then, as above, assuming you're going from a 70-200/4L IS to a 70-200/2.8L non-IS it only gives you one stop. Both those lenses are similar in price, so again it's a cost issue, not a performance issue. If you have the money the 70-200/2.8L IS is the best of both words.
If you are limited in budget to around $1050, then you do have the choice between the 70-200/4L IS and 70-200/2.8L non-IS. Which is best depends on what you do. The 70-200/4L IS is smaller and lighter and can be handheld at shutter speeds about 2 stops slower than the f2.8 non-IS version. On the other hand the f2.8 non-IS version is a stop faster, so when used wide open is better at stopping action because of the faster shutter speed the extra stop allows.
for more information on the 70-200 lenses