Your vet will need to perform an in person physical exam on the horse. S/he will run through the normal course of a physical (vitals, listen to lungs, gut, etc) to ensure the horse is healthy for travel.
If they choose to list the current vaccinations on the health cert, they can - most don't. You should have the original health cert and original coggins in the truck with you.
If you know you are stopping at horse hotels, you can provide them with a copy of the paperwork. I have a release for our horse hotel customers to sign.