I forgot to ask, are you using a full frame 6D/5D/1DX, or a crop sensor camera from the rest of Canon's range?
I'm assuming that he's using FF because a 20mm lens would give a 94 deg diagonal angle of view from his original post. And if he wants to do a whole room corner-to-corner, then a 17mm lens would get you to 93 deg for the horizontal AOV while a 20 would get you 84 deg.
To be honest, anything could work, even stitching. It is only a matter of convenience. TS-E may be the best choice (as a specialized lens). However, any decent gear combined with good post-processing skills could easily outweigh the best possible equipment, if you know what you're doing. 16-35/4L is a decent multipurpose option.
Stabilized UWA lenses are great for handheld video, but there's no way it could replace a decent tripod. At long exposures (like 0.5+sec) IS only compensates for vibrations, the movement stays + without a tripod it would be difficult to keep the perfect level alignment (if not impossible).
True, although horizontal and vertical tilts have their limits in software. Past 10 using LR, and the results start looking hokey. A TS-E is faster to use if a tripod is used. Shift can be used to compensate for tripod height changes and for getting yourself out of the frame when standing in front of small mirrors (i.e. small bathrooms).