Try to do - постарайся сделать. Try and do - что-то типа «сделай-ка».

This is only with TRY - we use TO, or AND in informal speech, but only in the future, conditional and imperative: “I’ll try and/to call him tomorrow.” -or- “Try and/to call him tomorrow!” -or- “If I had a phone I‘d try and/to call him.”

🙏

😉

So, for any other verbs, it means to be used as coordinate predicates?

I just don't understand how to translate it :(
E.g. 'Should go and vomit', can I translate it as "Меня сейчас стошнит"?

Yes, it’s a coordinating conjunction in other contexts. I don’t understand your example. “I should go and vomit” literally means идти куда-то И блевать

Thank you.
Oh, it's just phrase from film, it was said not literally. Watching public display of affection, and all that.