1.  Should I use construction sand or straw in the run?
I would use construction sand.  It drains well, is easy to clean, gives them something to dust bathe in, and they will use it for grit.  If straw gets and stays wet, it can get moldy.  If you put it in fairly thick, straw can pack together and be hard to clean out.
2.  I've read the food and water should be in the run, not the coop.  Is this true?  What if they get hungry/thirsty at night?
They cannot see at night.  They stay on roosts and don't eat or drink.  
Some of us provide food and water in the run, some in the coop, some provide it both places.  We all have different circumstances, set-ups, and management practices.  No one answer is right for everyone.  You can feed and water either place, whichever works out better for you.
3.  Will they be safe in the run while we are at work?  We have possums and coons in the neighborhood.
Probably.  With predators there are no guarantees, but that looks pretty good to me.  Raccoons are very good at unlocking locks.  I cannot tell what kind of lock you have on the doors and openings but it looks like they might be pretty easy to open.  You might consider a snap lock at least and preferably padlocks on all openings.  
4.  Is the roost too high for the pullets?
No.  They are fine.
5.  Do I put the grit in the feeeder or just lay it on the ground?  Do they get enough grit when they eat the sand/rocks on our paths?
I do not put grit in with their feeder.  I prefer them to decide how much they need.  If you use construction sand in the run and let them out to roam where they have access to the ground or your gravel walkways, they will get plenty of grit.
Good luck.  That should be nice for three hens.