How to say "Oops"; or other exclamation/interjection you might say if you make a mistake I was making an omelette and I dropped an egg on the floor.

I said "Oops!" because of my mistake.

Is Chinese "哎呀" (Āiyā) the nearest equivalent?

From what I gather, 哎呀 seems like a "sound word" -- by which I mean you may not see "哎" or "呀" used in other words - they are really used as exclamations or interjections, and they are very instinctive , knee-jerk reactions.

Are there other Chinese expressions someone might shout if they made a mistake?

To be honest, I probably said "shit" or "damn", so I wonder if there are a list of Chinese interjections , some mild ("oops!") and some more profane ("fuck!")

(Though I suppose it would be obvious if I saw someone drop an egg and shout in Chinese -- even if I never heard it before!)
There is no exact equivalent of the English "oops" in Chinese.

This is mostly because "oops" is a special English word, with an uncertain etymology but one theory is that it derives from "upsy-daisy". It is special because it is only used to express a mistake; I think you'll find that many other languages also don't have an exact equivalent of "oops".

Therefore I think you are right that 哎呀 is a close equivalent, but unlike "oops" it can be used to express general surprise, and it is onomatopoeic. Other related words that people can utter include the shorter "呀" or the longer "哎呀妈呀", depending on region and/or severity.

And yes, obviously there are many Chinese curse words that can be used to express "oops", similar to English words like "shit" or "damn". I won't list them because there are so many, and I'm sure many regional variants that I don't know of.