Yes, they are all part of mathematics. Placing a question mark at the end of a phrase does not make it a sensible question. Try to use a whole sentence to describe what it is that you want answered.
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Yes. It can also be negative in the numerator. Both positive and negative numbers (as well as zero) can be rational numbers. Both positive and negative numbers can be irrational numbers. Both positive and negative numbers (as well as zero) can be integers.
The quotient is what you get when you divide two numbers. If both numbers are positive, the quotient will be positive. If both numbers are negative, the quotient will be positive. If one number is positive and one number is negative, their quotient will be negative.
A negative times a positive is always a negative. The rest is the same: just multiply as if they are both positive numbers and then add the negative sign at the end.
Rational numbers can be negative or positive.
The product of all those numbers will always be a negative number.