It is always divisible by two.
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Yes, an even number plus an even number is always even. Also, and odd number plus an odd number is always even.
No. The opposite is true: the gcf of an odd number and an even number is always odd. All the factors of an odd number are odd; or to put it another way: If a number has an even factor, the number itself must be even. Thus the only common factors between odd and even numbers must be odd.
Two odd numbers always sum to an even number. Always. Two even numbers always sum to an even number, and an odd number and an even number always sum to an odd number.
The square of an even number is always even.
The sum of an odd and even number is always odd because there's always one left over. Think of any odd number as being any even number plus one. If you add two even numbers, the answer is always even. If you add an even and an odd (plus one), the answer is always odd.