Incomplete Answer, and probably not correct:
Three under one name, quick three times,
One has three letters. I feel like this cannot be the correct answer to this clue but it's the best I can do.
Once heralded as prime, I am still the first,
One is the first positive integer, and even though the only factors are one and itself, it is not prime. However, it was once considered a prime number.
I am deadly, and like no other,
Unsure why you are deadly, but you are like no other in that you are neither prime nor composite (except for zero, maybe that has something to do with this?)
Is it warm in here? Am I making you mad?
My first thought was absolute zero (which is the complete opposite of warm). Unsure of this