I don't see any "set of actions to use on the environment" in the King Quest screenshot. However, I do remember certain adventure games having a menu below with somewhat large list of available actions, like in Indiana Jones series:
Personally, I don't like these kinds of menus much, especially since for most interactible elements only a handful of these actions actually make sense.
In that regard, I prefer simplicity. Depending on the kind of game and the kinds of interactions, I could use one of the following approaches:
1. Simplest: clicking an environment item has the player acting upon that item.
Depending on the game, the player may also select one of other items from inventory to use on environment item, or even combine two inventory items together.
This is - in terms of number of available actions - vaguely similar to this "give/pick up/open/close/push/pull/etc." menu.
However, it's slightly different, because there's always the default general-purpose action you can take. Not only that, the general-purpose action can give you a hint about which item should be used. E.g. when interacting with a lamp, you could get something like "if only I had something to ignite this lamp".
2. A wee bit more complex: similar to 1, except you have "Examine" and "Act" actions separately (maybe assigned to different mouse buttons, or maybe selectable from the menu).
Might make sense when "Act" can cause some regrettable and undoable action, forcing/encouraging the player to load the game. Though I suppose if an action is regrettable, a yes/no question might be more appropriate.
3. When acting upon an environment item, have a contextual menu pop up with actions available; these actions making sense for the given item (e.g. push left/push right when used on a cupboard). Is it how Gemini Rue works?
When there's always only one action that makes sense for a given item, you might want to skip the menu for this specific item (in fact, such items might be a majority).
Personally, I'd try aim for 1, or maybe if I have a mood for writing witty examination descriptions, for 2.
I could use the contextual menu occasionally, especially if the environment element has multiple possible desirable actions. I'd probably be more eager to use that option if I had setup a dialogue system with choices in the first place - then I would just apply the dialogue system here. After all, mechanically speaking, environment elements aren't so much different from NPCs you could encounter.
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As for the pixel hunting - I generally would avoid it.
One feature I've seen in several adventure games is ability to reveal positions of all interactible items (e.g. with some kind of ring appearing around the item in question). This avoids having all items be super-visible all the time, while still preventing the player from getting stuck because they missed an interactible element.
Not only that, I'd avoid items that have only 1 pixel of clickable area - even if item itself appears teeny tiny, I'd give it a little extra clickable area around it, so that the player doesn't need to move the cursor at the exact position.