I will happily do what I can to help you.
My girlfriend has a nice leg.
(from descriptive homework)
First of all, you can say "a" only when you can say "one". I don't know it for sure, but I was told that in old English language there was only one word for "a" and "one", something like in French language, if I am not mistaken.
As the result, first of all, you can put "a" only before countable singular nouns.
He wants to become a dancer.
They want to become dancers.
In the second sentence "a" disappears, because we cannot put it before "dancers".
So if you say "My girlfriend has a nice leg", it is pretty much the same as saying "My girlfriend has one nice leg".
There is a little difference between "a" and "one". In usual sentences, or so called "unmarked" sentences, we normally use "a" or "an". But when it is important to stress "one", not "two", "three" etc, we use "one".
My friend has a Mercedes.
My friend is very rich. He has one Mercedes, two Jaguars, and three Rolls-Royces.
By the way, you can use "the", definite article, before uncountable, countable singular, and countable plural nouns.
In summer the kids (our kids) always go to a camp.
Ouh, the whiskey is horrible!
Okay, I must get to work now, but I will answer all the other questions shortly.
Have a good day!