What is che ti doing in there? I saw nothing that would make me think i am taking action here, like io, mi, me. I'm lost.
Yeah it definitely needs congiuntivo here; switching congiuntivo with presente is something you hear pretty often in colloquial speech though (wrong nonetheless).
"Do you want that I take you home?" Is it possible on that way? Somebody please explain that.
To me it sounds like what a New Yorker might say, but not technically correct English.
You're right about it sounding rather like New York and/or maybe New Jersey dialect. I wouldn't say it isn't correct - especially not while I'm standing on a street corner in Yonkers - but it's probably safe to say that it isn't standard broadcast English.
From Massachusetts, this is standard here too. In the context, for example, your friend is feeling ill at a party you'd ask, "do you want me to take you home?"
It's not how we say it but it's a literal halfway between the translations. Most people would understand if you said that to them.
Would "Do you want me to bring you to your house?" be a suitable translation? I know there's no literal possessive there but I've seen many duolingo sentences express an implied possessive.