Yes, "tá" is the short for "está", it is not really a word, but you will not hear anyone in the streets saying "Está, desculpa". The short form is used when someone wants to express agreement. Example: Você quer ir ao cinema? (Do you want to go to the movies?), reply Tá, vamos (Ok, let's).

The test asked me to choose between the answers "Tá, desculpa." and "Tá, desculpo."

I chose the former since it's the most common one, but both are correct (albeit they have different meanings). If someone asks you "Você me desculpa?", you can answer with "Tá, desculpo."

I'm reporting that "alright" should be accepted as an alternative to OK.

I have never heard of this word. Is it a shortening of "está"?