The first is correct along with the third. However, "I am" means you are for sure you are and "I feel" means you are unsure if you are or aren't.
The second is different because "I am cold" is basically "My body is cold." and "I have a cold." means you are sick.
"A cold" is a sickness and when you have one it doesn't mean you're cold, it just means you're sick.
Hope this helps^^ Have a nice day!