Its problem is not grammatical, but idiomatic. We say take an exam, or sit for an exam, but we do not say attend an exam. A sentence may be grammatically correct and still wrong.
So you can learn that it is grammatically correct to say exams, not, as you put it, exam
To have grammatically correct and pronounce the word very well.
yes
'She does not have her book' is grammatically correct.
He, you and I is not grammatically correct. The proper way to say this is you, him and I.
No, it's having a meeting or attending a meeting (or you might be holding a meeting if you're the boss).
Yes, soonest is correct grammatically.
"Not like that" can be grammatically correct, depending on the context.
Yes! That is grammatically correct!
Yes, 'for free', is grammatically correct.
Yes it is a grammatically correct
Currently we have is grammatically correct.