Yes full frame lenses work just fine on crop frame bodies. It's just that you aren't using all of the glass / it is heavier or more expensive than it would otherwise need to be to cover your smaller sensor.
That said, Canon doesn't happen to make any crop-frame lenses that have their top of the line professional grade optics in them. So while in theory you could have the best of the best glass in both full and crop format versions for efficient weight and money, and while that would be great, as it so happens, all of Canon's best lenses are actually full frame ones. And in fact, they make very few full frame lenses that are of questionably quality at all. So generally, full frame lenses tend to be upgrades.
Regarding the 24-105 f/4L, it is a fantastic lens, with near perfect optics and good autofocus and build quality. However, I just recently sold mine and replaced it with a 28-135 f/3.5-5.6 (also full frame, not "L") lens. So far I am happy with my decision.
1) The 28-135 costs 3x less. I sold my 24-105 used for $630, and got the other one used for $220.
2) The optics are nearly as good, and the aperature on average nearly the same (faster wide, slower tele, same mid)
3) I take more walkaround photos at the tele end than the wide end, and 28-135 has more useful reach.
4) It weighs about 20% less which makes me slightly happier to walk around with it.
Mainly the cost, though. With those extra $400 I can buy other lenses. In my case I bought two lenses:
* An 80mm 1.4 for portraits
* a pentacon six medium format lens + a pentacon six to canon tilt adapter = cheap legitimate quality sharp tilt lens, for portraits, still life/product, landscape, artsy fartsy stuff.
AND that was with me getting it cheap as a kit lens with the 6D. If you were to buy it new, you would be getting an outrageously overpriced lens for what it does compared to the alternative cheaper option, IMO. Used price is still high I think as discussed above, but if you care more about perfect image quality and uber sharpness than you do lots of options, then the 24-105 might be the better choice for YOU.
Depends where your priorities lie. Or if you just have a lot of money then by all means get the 24-105 as well as whatever else you want.