What player actions can be specifically tracked by command blocks?

Using commands you can check for everything that is tracked by the scoreboard (see https://minecraft.gamepedia.com/Scoreboard#Criteria) or trackable through advancement triggers (https://minecraft.gamepedia.com/Advancements#List_of_triggers).

sadly, none of the things you mentioned (open door, step on pressure plate, click button) are trackable at this point.
(which is also why mapmakers often use clickable signs instead of buttons so they know which player clicked it)

Moving my comment to an answer:

The format used for storing text on signs allows you to embed "onClick" events, including commands that will be run when the sign is right-clicked. The command will target the player that clicked as @s, and will be run at the coordinates of the sign. Very useful, and seems to be just what you're looking for.

Here's a nice generator: minecraft.tools/en/sign.php

You can find more information on sign's data format here, and on Minecraft's JSON text format here.