OK, a few things first. "Slang" is uncountable, so we never say "a slang". That's a learner error. Also, proverbs are not slang. There's a huge difference. As far as I can tell, it's not really a standard phrase in English, but Chinese websites for learning English say that it is. Some sources say it's Irish, some Scotch, even one says it's Romanian. My point is, don't feel a need to learn it and use it - we'd probably think you're translating from Chinese. The phrase sounds like it has an innuendo, and that'd be my first thought as well. However, apparently it means "don't interfere in someone else's business."

As far as I know, it's not a common saying, but I suspect it means don't kiss( or do anything similar with) another man's girlfriend/wife.

I don't know the context but I think that this phrase would be pertinent when you don't want somebody to take or to use something of yours. Hard to say more concrete without the context.

Don't commit adultery. =)

It is not 'slang' more like an old-fashioned proverb - yanyu 言语 of Irish origin. It means 'Mind your own business!' 关自己的事。