Only as you move to reframe the subject to the same size. It's the movement that makes the difference. Leave the camera behind & compare the look of something close up to the same subject at a distance & you'll see close up distorts the perspective due to the relative distances being so different. Stand ~10' from someone & their nose & ears are both ~10' away (close enough not to be noticeable) then stand 1' from someone & the roughly 6" distance from nose to ears beomces significant, the ears being 1.5x further from the lens look smaller.
No, - it's image plane (at infinity) to the lens. The focal point is where the lens axis meets the image plane when focused at infinity or perhaps more accurately the point where a light source on the lens axis at infinite distance is brought to focus (which means the same thing). With simple thin lenses this defintion was easy, but with most photographic lenses we don't know which point on the lens the measurement needs to be made to. I believe it's the 'rear principle point' which doesn't have to be within the physical lens at all - just as well or focal lengths below ~40mm would be impossible on SLRs. See Focal length - Wikipedia
I'm currently having a right good look at the Samyang XP 85mm f/1.2, which seems to equal and even out do the Sigma 85mm Art f/1.4 and the Canon 85mm f/1.2L ii in some respects and at around 1/3 of the price of the latter. It's MF but certainly the reviews I've seen say it's an astounding piece of glass.
No, the huge difference is due to perspective from different shooting distances. If you shoot both lenses at the same distance, then crop the image from the 24mm to match the 105, you will have the same image, assuming the resolution is still there after the crop. The distortion comes from subject to camera distance, i.e. perspective, and NOT from the focal length of the lens. When you said "at the same size," you forced a change in the camera to subject distance, which is the cause of the perspective change. The 24mm lens did not cause the exaggeration of facial features, moving closer caused that. EDIT: I now see that Petrochemist already answered this. Oopsie!
Isn't this what I said? You would have to move the camera and make the subject the same size in both images.
But it's moving the camera that changed the perspective, not the different focal length. The focal length accommodates the camera position. Shoot an image at 105mm, shoot another at 24mm, without moving the camera, then crop the 24mm image to match the 105. They will be the same, because the camera hasn't moved. To get the different lenses to show the same size image, you do have to move the camera, but moving the camera is what changes the perspective, not using a different lens. Shooting both at the same distance and cropping the one to match the other will yield the same image (assuming you have enough resolution for the crop.) It's the distance, not the lens, that makes the change in the images.
Like my signature says, shoot loose and crop. Many people don't realize that cropping a picture with from a 50mm will give you the same perspective as an un-cropped tele.
That depends on your camera. With 20-24mp being the norm these days, that isn't as much of an issue as it use to be.
Judging by some of the cropped images I see, it's still a problem on digital camera's and often overdone, rendering on occasion, some quite horrendous IQ. Sometimes less is more.