I never used to cross my zeros until I was spending some time writing a lot of diagrams that had both O (for oxygen) and 0 (for the numeric label). It got very confusing!

I started crossing the Z in college because I was tired of messing things up because I confused a Z for a 2. I only crossed the 0 when Os showed up and it could be confusing. Similarly my 1s would get the hat and feet if I felt like it wasn't clear what it was.

I can't say I ever had trouble with y and 4 looking the same. I use the open 4 but the y has one less angle and it sits lower. If anything a y is more likely to look like a v if you can't see the descender clearly

I always cross my zs in maths to distinguish them from 2s, but in some countries they cross the 2s to distinguish them from zs, so the ambiguities remain.

I started crossing Z, 7, 0 and generally writing in as unambiguous a manner as possible in my math homework because I was afraid of the teacher reading it wrong and taking points off.

To avoid mixing up zeroes with the empty set or phi, simply put a dot inside of them instead of crossing them. Many fonts do that.