It depends on context and which kind of like is meant. Both of the following are correct:
"Mary likes me" and "Why would she care for a man like me?"
"Not like that" can be grammatically correct, depending on the context.
No, "I like him best" is grammatically correct. It is a simple sentence in which "I" is the subject, "like" is the verb, and "him best" is the object.
The phrase "Is you don't miss me do you" is not grammatically correct. It should be rephrased to something like "Don't you miss me?" to be correct.
It can be, like in the following sentence. There were many firsts in this historic week.
Yes! That is grammatically correct!
Yes, 'for free', is grammatically correct.
I like the kind of person you are.
This is not grammatically correct. The correct form is 'you do not know', or the abbreviated 'you don't know'.
'What a drunkard you are' is a grammatically correct English sentence.
"That was wrong" is a grammatically correct sentence.
Yes, Almighty God is grammatically correct.
Yes, it is grammatically correct.