Yes, the cans are new construction type and most are the 6" size ones. I can't remember how much the ceiling stuff was, install was around 6 years ago and I'm sure prices are different now anyways. Just know that this type of system doesn't allow for just an easy one tile pop-out like a conventional system does. If a tile 4 tiles in from the wall needs to be removed then all 4 tiles would be removed. It's not a system that you constantly want to access above but it does allow for it if necessary.
In your situation with no piping above your ceiling that could leak you may be ok to just drywall and have an access port for your spigot. I had a spigot iso valve that I needed to access and it became a pain to take down a tile constantly to do so. I eventually moved the iso valve to an unfinished side of the basement so not to worry about it.
If you do drywall, your idea of installing remodel cans may be your best way to go to make sure the holes are exact. Since the recomendation for ceiling drywall is 5/8" drywall due to the sag factor be aware that there is special ceiling drywall that's 1/2" that you can install instead of the 5/8". It's made for ceilings and stronger than regular 1/2". Would be alot easier to install than the hevier 5/8".
And I thought I'd provide one more link for you on recessed cans installation that might help with a few more details if needed.
http://18004654533.com/Howto/Install...edLighting.asp