Last time I'm aware the monarch did something, it was the firing of the entire Australian parliament and calling a snap election. Which is quite a sensible thing to do if parliament is completely jammed, and since it was reasonably sensible and popular in the situation it happened, nothing happened to the monarch. (Imagine if someone had the power to do that to Congress! Maybe they'd actually do stuff again, and we'd have to rename them to Progress)

This is an alright use of a monarch IMO.

Yes. This fits with the UK monarch's modern role which is to ensure continuity of government.

The UK example would be if the UK government lost a vote of confidence, the monarch might ask the opposition parties to see if they could form a government that could pass a vote of confidence to avoid a general election. Although parliamentary math makes such a scenario unlikely in practice.

On the UK end I don't think the monarch can sack the Prime Minister. While the UK monarch could strongly suggest to a Prime Minister that it's time to resign. Usually when a Prime Minister gets to that point, it's usually the end of the road and they won't last much longer anyway even if they pushed forward.

The monarch has a lot of soft power which under the modern system is usually deployed to ensure continuity of government rather than in a controlling manner.

(The Australia, Canada, etc constitutions gives the monarch different powers than the UK and it's usually delegated to whoever is the governor for that nation)