There are several questions to unpack in your query, so I'll address them individually.
if I put multiple filters on a camera, does that degrade image quality a lot more than just one filter?
The link to Roger Cicala's blog post in Bart Arondson's comment sprang immediately to mind. Yes, stacking filters will cumulatively have an effect on image quality, but noticeably less so with good filters.
Is it effected by the filter type (UV, Neutral Density or Circular Polarizer) mounted?
Mostly the effect is mitigated by the quality multicoatings on better lenses, rather than by the type of filter (UV vs ND vs polarizer). I doubt you'll find anybody who recommends stacking a UV filter with any other filter. (UV filters are always a hot debate in any forum anyways). There are many cases where you might want to stack a polarizer with one or more ND filter.
Is image quality effected by the filter mount type (e.g. screw-in v square/rectangle ones) or is it simply down to the price?
In terms of light leaking between the filters, screw-in filters are better in this regard. With square filter holders, and especially with high-ND filters, you have to be careful to wrap or shade the portions of the filter outside the image circle to prevent "external" light bouncing in between the filters, or between the filter and front element. This is essentially not a problem with screw-in filters.
If you are going to stack multiple filters, vignetting becomes an issue with screw-in filters much faster than with square holders. Assuming you standardize the size of your screw-in filters, if you want to make sure you don't vignette on a particular lens, you would conservatively have to greatly oversize your filters much larger than you typically would choose for your lens/camera system.
If you choose to ever use ND grad filters, the only reasonable choice is to use square holders. There are screw-in ND grad filters, but you cannot shift their "horizon" line up or down, so screw-in ND grads completely remove the vertical axis from your choice in composition. Conversely, you can position your rectangular ND grad however high or low you desire in order to suit your composition's horizon. This is especially important if you cannot recompose if you are taking multiple shots of the same scene, where you need to change your filters during the shot sequence.
So indeed, there are important considerations other than price when it comes to choosing between screw-in and filter holder systems.
In the Context section before your questions, you wondered,
but I'm curious if some people, rather than buy a Variable Neutral Density, buy all Neutral Density Filters from 1-10 stops and select the one they want to use, or just stack them to get the correct value they desire.
Variable NDs are okay for photography. Realize that a variable ND is essentially just 2 polarizers mounted in a common body, and they can rotate with respect to each other. Polarizers in general do not work well on wide angle lenses, because the polarization axis changes over the width of the scene with respect to the lighting in the scene. This usually results in variable color of the sky, or in the case of variable ND filters, an "X" pattern of changing ND amount. But, if you are using normal (or longer) focal length lenses, both of these problems are reduced.
Variable NDs, as well as a full range of ND filters, are often used when shooting video, as they give back to the videographer another factor of control in their compositions. Shutter speed is not under the cinematographer's control, so light control and ND filters are heavily used to control the exposure in scenes.
Personally, when it comes to ND filters in DSLR photography, I have never needed to have every single stop from 1-10 as an option. I can easily find an additional stop of reduction by combining, say, a 1/2-stop smaller aperture with 1/2 stop slower ISO. I use 1, 2, 6, 10, and 16-stop 100mm square ND filters. I have no problem with stacking two of them to get the effect I desire, and often put a 105mm circular polarizer in front of them as well.