No.
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Yes, -3 is both a rational number and an integer.
-16 is both a negative integer and a rational number
By definition, an irrational number is a number that is not rational, or in other words a number that cannot be expressed as an integer divided by another integer. A number cannot be both "rational" and "not rational."
Integers are whole numbers, with no fraction or decimal part. Rational numbers are numbers that can be expressed in the form a/b where a and b are both integers. Thus 1/2 is a rational number, but not an integer. 0.34343434... is a rational number (as it can be expressed as 34/99) but is not an integer.
All integers are rational numbers.