Ideally, the window jamb should be flush or slightly proud of the interior wall finish so that when the window casings are installed, there is no gap between the casing and the wall. The window jamb width needs to be equal to the total wall thickness (stud + exterior sheathing + interior drywall) for this to work out right. If the windows jambs are not the right width, there is probably little that can be done about that problem, since you most likely can't trim a vinyl jamb. (It's easy to do this with older wood windows.)
If that's the case, one thing that can be done is to cut a rabbet in the back of the window casing where it meets the jamb to account for the difference in how far the jamb protrutes from the interior wall surface. Of course, this only works if the difference is small enough and the casing is thick enough, which sounds like it is not the situation in your case. And in any case, this is something that a typical replacement window installer cannot afford to spend time on, unless you are paying a premium price for excellent work (Our neighbors paid ~$1K per window for replacement windows in an old house that required extensive amounts of this sort of attention to detail.)
I can't see the windows from here, but my guess is that you are stuck with the gap and just need to fill it. Caulk is your friend in this case, and is fairly common thing to do to fill smallish gaps, although
1/4" sounds like a greater than average gap to me. Don't fill the gap with drywall mud, that will eventually crack and show the seam. That is unless they are willing to put on a 1/4" thickness or whatever is required, and then feather it out across the wall to bring up the whole wall surface flush with the back of the casing. (And which you will obviously need to repaint a large area.)The "stool" is the flat horizontal piece at the bottom on the inside, which is commonly and incorrectly called the sill, which is what I assume you are talking about. The stool should be nailed such that it is solid, so I would complain if it moves when you press on it. The trim piece that goes flat against the wall underneath the stool is called an apron, which in addition to provinding a finished look to the window, also provides some amount of support for the stool. Was this installed?
Call back your installer, and use words like stool and apron, that will make him think you know what you are talking about. Perhaps that will help convice him that he has a bit more work to do.
Ken