If a tenant gives thirty day notice (pays for the month) but surrenders the keys with say 20 days remaining can I start construction early?

As said, once he gives you back the keys, then he has surrendered possession. Make sure ALL his effects are removed before you start work however. I'd do a walk-through with him so both sides are happy! Whether you give him back more than his normal deposit, re rent already paid over, is up to you, but legally you are not bound to do that. His decision to move out early.

I would say that since he gave you the keys, he has relinquished his rights to the rental unit. If he wanted to retain access to the unit, he should have retained possession of the keys. You don't have to return 20 days worth of rent (unless you want to). But the problem might occur if he has a duplicate set of keys (which I doubt). He could come back and say he never gave you his set of keys. Did you issue a receipt acknowledging the return of the keys?

If the tenant turned in the keys then they have left and you can enter

Once he give back the keys, he surrenders possession of the property. You are free to do as you please with the property. Remember, it was his choice to vacate earlier than the full 30 days, so you are not required to return any unused rent.

Once he turns in the keys, you can start.

sure....just give 24 hours notice before you enter....you know the drill...the law doesn't change..otherwise it would say so in the law

if he's gone and cleaned out, sure

you can start now.