Zero is not defined to be even. A number is even if it's twice some integer. Zero is twice an integer (namely 0), so it's even.

If you try to bring personal intuition into mathematics, you'll just repeatedly shoot yourself in the foot

I disagree. Intuition should always be foremost in mathematical understanding. You may have gotten a different impression from your math classes, and that IMO is a consequence of bad teaching.

I had a similar problem with accepting that 1/inf = 0

As you should; the statement is false. The limit of 1/x as x->infinity is zero, but there is no x at which the quotient "reaches" zero. In non-standard analysis, if x is positive infinite, then 1/x is positive infinitesimal, meaning it is infinitely close to, but still greater than zero.