I've heard that EF-S lenses are not compatible with full frame bodies and that using such lenses could damage the mirror.
Are the EF-S lenses not compatible with full frame bodies in terms of "not optimal image quality with larger sensors" or are there more important problems like damage of equipment? Does the same issues apply for both fixed lenses and zoom lenses?
The main problem with using EF-S mount lenses on a FF camera is as you said the risk of damaging the mirror. EF-S lenses can protrude further back into the camera body than EF lenses, which means the mirror might hit the rear of the lense. This is what can cause damage to the mirror itself or the mechanism for flipping the mirror.
For some EF-S mount lenses this might not be a problem, since the lens doesn't extend back far enough to cause the mirror to hit it. Depending on the lens design and if the lens is rear focusing or not. Rear focusing lenses will use the rear group or elements to focus and this can extend the lens backwards into the camera. These lenses can work on a full frame camera at some focus distances, but not others. A zoom lens can also use the same technique when zooming, and cause the same problem.
The answer is really that it depends on the lens if it's safe to use or not.
The image quality will not be as good either, since the EF-s mount lenses will be optimized for smaller image circles than EF mount lenses.
EF-S lenses manufactured by Canon have an extra tab that prevents them from being mounted on Full Frame and APS-H Canon bodies. Many "digital" lenses made by third party manufacturers do not. You should be careful when using them on full frame bodies because the full frame mirror passes much closer to the lens mount flange than the mirror on crop bodies does.
For example, I can mount a Tamron AF 17-50mm f/2.8 XR Di-II LD SP Aspherical (IF) Zoom Lens to my Canon 5D mk II. The image circle does not cover the entire sensor. It does have a large enough image circle to reach from bottom to top in the center of the frame. You could crop the image down to the pixels contained in a slightly larger than APS-C sized area of the center of the sensor. On the 5616X3744 (21MP) EOS 5DII at f/3.2 this yields slight corner darkening (blur from the OOF edge of the circle) at about 3900X2600 (10MP) if you want to maintain the 3:2 ratio. You can also crop to 2792X3744 (10.4MP) in either 3:4 or 4:3 orientation. Or 3200X3200 (10.2MP). This lens does NOT have the tab like the Canon EF-S lenses do that prevent you from attaching them to a FF body. Auto focus, metering, and aperture operation work as expected.
The best way to check a third party lens is to look at it when not mounted to a camera body. Run the zoom from short to long and back with the focus at infinity. Repeat with the focus at Minimum Focus Distance. With the lens at the widest focal length, run the focus from MFD to infinity and back. Repeat with the lens at longest focal length. If no lens elements protrude past the flange on the rear of the lens at any time you are probably safe that the lens will not damage your mirror.
The back element of EF-S lenses can be closer to the sensor (but it doesn't have to be) and it can actually hit the mirror and damage the lens and/or body.
Canon EF-S lenses are mechanically incompatible with EF mount by design. That said, many of "crop" 3rd party lenses have EF mount instead.
For 3rd party lenses some take advantage of the extra space at the back and can hit the mirror on full frame cameras and some are just "optimized for" crop sensors.
Note that the back element can move when the lens zooms or focuses so it may work sometimes.
I don't know how dangerous it is to use the Sigma 105mm but I wouldn't risk it on a new camera unless I know this lens is safe.
