#1 - Lizard diseases aren't airborne. As long as this person doesn't go around mixing food dishes and water bowls across all the lizard tanks, they aren't going to catch anything. They would just need to wash their hands before and after when taking care of them, which is what you should be doing between lizards anyway.
#2 - You're thinking about bird eggs, not geckos.
#3 - That's a matter of opinion.
Hence why it said "not positive".