About the Master Cave Spider's Web ability, the CRRG says:

Web
Card text: “Web: Each hero adjacent to this monster must suffer 1 stamina to move out of his current space; this is in addition to any other stamina suffered to move.”
• A hero at full stamina suffers 1 health if he moves (or is moved) out of a space adjacent to a monster with Web.

Do heroes suffer 1 stamina if they move out of such a space with an ability that doesn't use the word "move", such as Astarra's hero ability ( Spend 1 movement point to remove your figure from the map... ) or the skill Shadow Step ( Exhaust this card to place your hero figure in a space that contains only a Shadow Soul ), or do they not?

No. Remove and place effects do not trigger the web. Check the CRRG entry "movement":

When a figure changes its position due to an effect explicitly
including the keyword “move“, it is moving out of its current
space and moving into another space. A figure being “placed” is
not moving into a space.

Good thanks. I actually checked this entry, but I was made unsure because of the entry just above, under Entering and Exiting a space:
• When a figure changes its position from one space on the map to another space on the map, it is considered to be exiting its current space [...]

I wondered if "exiting" meant "moving out of". Maybe it would be good to add, under "Movement", after A figure being “placed” is not moving into a space : A figure being "removed" is not moving out of a space.

Alright. Will look into this.