Hi Wendy! Here are my answers to your questions:
I've got no choices. = "I don't have any choices," and "I don't have a choice," are correct.
Sometimes, you may hear people use the expression, "I've got no choice." It's used to put emphasis on "no." It's never used with the plural "choices."
I had no choices. = Similarly to the phrase above, "I didn't have any choices," and "I didn't have a choice," are correct.
However, you may hear people say, "I had no choice." It's used to put emphasis on "no." It's never used with the plural "choices."
He leaves me no choices. = "He leaves me no choice."
He left me no choices. = "He left me no choice."
I have heard people saying that I've got no choice? ( Should we use choices or choice?) = "I've got no choice." We only use the singular form for this expression.
Hope that helps!