I don't know what kind of company you work for, but, in spite of the fact that every single technical interview I've ever had involved live coding, I've literally never had to write code in front of anyone at work. By "write code" I mean "write code that's expected to compile and run." I've done plenty of whiteboarding at work, but never anything like what happens in a technical interview.

Of course. Anyone who lacks the ability to complete the assignment can submit a solution written by someone else and be told how it works. Where they’ll fail in follow-up questioning is when you dive into their decision making process.

I have only found one person to have cheated in an interview, and they were exceedingly easy to identify as a fraud when questioned in person.