If I book a hotel room five nights but end up staying only four will they refund the extra day ??

pizza, you are wrong. It is Northwest

Are you on southwest?? becuse if you are not then they will give you no refund 

I travel for both pleasure and work a lot, and quite often leave early.  Generally, payment isn't processed until checkout. I can usually give notice early on, but even on those occasions when I've told them I plan on checking out a day early only one day in advance, they haven't charged me for the night I didn't stay.  That said, I never pay more than one night's deposit in advance, and stay at decent hotels.  It may be different if you pay for the entire stay up front, or stay in a cheap hotel. 

Definitely not if you have to pay for the whole stay in advance. 

If you can make an ordinary reservation and leave before the 5th night they generally won't charge you for the last night as long as you check out before the deadline.

you may, depends how polite and courteous you are with the front office.

What does that have to do with Books and Authors? If you're writing a book about that, then call a hotel and ask them.

Only if you write a novel about the hotel before you leave.

That IS why you posted this to Books & Authors, rather than Business & Finance, right ?

Not always, check with the hotel as different hotels will have different rules.

That depends on the conditions under which you book the five nights.  Was it with or without a refund?  Was it a special lower rate due to the length of stay?  When did you cancel the last night--can the hotel resell the room in time?

You can ask but I think not, certainly not the big corporate ones but an independent might, they don't tend to have a head office dictating to them about what the policy is for them.