At least in HL2 there was no way, short of starting a new game session or aliasing a command to something else. Maybe you can alias it to itself?
Fairly easy, I assume. Looking at the Stylechanger, that version of VBSP replaces brush entities based on their textures and nearby instances, so it should be possible.
There are settings to allow placement on gratings, goo, or glass, which I think are all set to True for the triggers, but not other testing elements. But there's still no feasible way to put it in midair.
Regarding the door antline attachments: What I have come to realize is that the door is essentially a lightstrip: from the viewpoint of the editor, it is a flat element on the floor, and all antline attachments to it must be parallel to the element, i.e. flat on the floor. This means that you can put four signages on the floor in front of the door and four signages on the floor behind the door (and probably, one to the either side of the door, still on the floor). The floor surfaces get bumped onto a wall if there is no floor at that position. I have edited the antline attachments to only keep the four positions right in front of the door, since the positions behind the door are rather useless and usually only confusing.
Is there any way to change this situation, so that you could have signages on the sides of the door, much like with the exit doors: four on top, four on bottom (the top and bottom ones get bumped to the floor if there is no wall surface), four on left up-to-down, four on right up-to-down? Or would any such solution involve having to anchor the entrance of the door against a wall?
No, because the door is like an object in the middle of the floor. Not on the wall.
No other item in the PTI has antline connection points aimed at an area not part of it's base. I don't know if the PTI is even capable of handling wires to an area above the square. It might be fixable though if a new item was created that is placed upside-down on the top of the block. The antlines would be wired into the front, with four attachments there to trick the editor into placing indiators above the door. Then the item could have a start-reversed option that inverts the door, making it place like it is now. I'll have to go test this to see if it works.
But since there isn't actually a wall there it would be placed on the floor, bringing back the same problem we have now.