Can I use the discounted movie tickets from Costco here? If yes, will there be a surcharge and will I be able to buy tickets a week in advance?

Excellent

If they are "premiere" passes, then they will work at any Regal owned location. You can use them for any show time currently on sale to the public for a normal showing (excludes things like fathom events). Any 3d, RPX or IMAX showing will have a surcharge. They can't be used online and the physical passes must be present at the box office to exchange for a ticket