Depends on the interview or the level but I prefer open-ended situations like "imagine a local small business that sells widgets. They want a backend that keeps track of inventory and a frontend that displays widgets and prices to customers." Or whatever the idea is. Some candidates only need that much and begin asking questions, designing systems, writing code, whatever they want. Others need more guidance and I'm happy to continue defining how this app should work or how big the audience will be or any other details. To me it's about exploring their experience with the whole process holistically and letting them dive into their subject matter expertise naturally. From this idea, there is no "completion" of the exercise as no one is actually going to write a backend and frontend in 45 minutes.

But I think you learn an enormous amount about an engineer here. How will they use this time? What will they focus on? The details they choose to spend time on, how did they turn out?

I've seen Principals take the whole 45 minutes on system design asking extremely detailed questions and building a system far superior to anything I would have engineered myself (interviewing up level). I've seen Jrs hash out a simple database structure in 5 minutes in a comment block and spend 40 minutes writing a simple react app that shows this hypothetical data. Obviously this principal wasn't hired for their front-end knowledge and this Jr wasn't hired for their system design, but still, very valuable to see how they do it.

Do you provide feedback to rejected folks so they understand what they should try to improve if they re-apply? The lack of feedback from most places completely puts me off from considering them again, even if that’s what they do to everyone.

I am willing to provide a short communication some times providing information but the reality is usually a lot less interesting than people think.

Reality: I want to hire 1 person, we get say 5 past all the stuff before me, we interview all 5, and then we choose the one that we think fits the best culturally, personality wise, skill wise, etc. So our reply is usually "hey you were great, we wouldn't interview you if you weren't, we just found someone we liked a little bit more". Which isn't very satisfying but you have to think about it from my perspective too. How do you choose between 5 great candidates?

Ultimately, luck is playing a massive role here https://www.youtube.com/watch?v=3LopI4YeC4I