I stood on my cat's tail and she has been avoiding me for two days — is there any way to actually apologise to a cat?

She has formed an association, not a grudge, and that distinction is the whole answer.

What happened is: your presence, in that context, preceded sudden pain. Cats learn that pairing extremely fast — it is a survival mechanism and it does not require any judgement about intent. She is not deciding you are careless. She has a prediction that being near you might hurt, and she is acting on it.

The good news in that framing: an association is undone the same way it was made, by new evidence. Every time she is near you and nothing bad happens, the prediction weakens. Time alone does some of this, but only if the encounters actually happen — a cat who avoids you successfully for two weeks gets no new evidence at all, which is why "give her space" taken too far can stall the repair rather than help it.

So the target is: frequent, unremarkable, non-threatening proximity.

What actively helps, in rough order of usefulness:

Be in the room and ignore her. Sit down, read something, do not look at her. Cats find direct attention from a large animal intense at the best of times, and staring is a threat signal. Being boring and present is the single most effective thing you can do.

Let her set the distance. Never approach; let her close the gap. If she gets within a metre and stops, that is progress and the correct response is to keep doing nothing.

Take over the food. Being the source of the meal is a fast, uncomplicated re-association, and it happens twice a day without you having to engineer anything.

Slow blinks. When she does look at you, close your eyes slowly and open them. It reads as non-threatening and a lot of cats return it.

All of these are about lowering the stakes of being near you. None of them involve making it up to her, because there is nothing to make up.

For question 3 — the things that reliably make it worse, all of which are what a guilty human wants to do:

Following her. Going to find her, crouching down, reaching out. From her side this is the frightening thing pursuing her, which confirms exactly the prediction you are trying to disprove.

Picking her up. Removes her ability to leave, which is the one thing she is currently relying on.

Treats offered from the hand, too early. The food is good and the hand is the problem; forcing them together while she is still wary can attach the wariness to the food. Put the treat down and walk away instead.

Apologising out loud, at length, in a concerned voice. Tone carries. An anxious, insistent voice directed at a cat is not reassuring, it is more input from the thing she is avoiding.

Two days is short. Most of these repair within a week if nothing pushes.