Any vehicle brought to Washington from out of state has to be inspected by the State Patrol before a WA title can be issued, period. It's not a big deal-- usually fifteen minutes or so. It is true this is a VIN check: physically verifying that the VIN on the title matches the one(s) on the chassis and running the VIN against a nationwide database to make sure the vehicle was never reported stolen.
And while emission testing is not required state-wide (yet), it has been required for at least twenty years in the areas surrounding the larger metropolitan areas in the state (Portland, Seattle/Tacoma and Spokane). A 'pass' is required every other year as a condition of renewing license tabs. Gig Harbor is right outside Tacoma, so they probably do require emissions testing, but bear in mind cars over 25 years are exempted outright anyway.
As to the designation of year and make: looking at Washington law, per RCW 46.04.251 it's definitely a 'kit vehicle'. According to RCW 46.16.680, when a WA title is issued, the make will be designated as 'KITV' (for kit vehicle), and the model designation must reflect the year and make the replica emulates, plus the word 'replica'-- e.g., '57 Porsche replica'.
As to the assigned year for the vehicle: "For a manufactured new vehicle kit and a manufactured body kit, the model year of the vehicle is the year reflected on the manufacturer's certificate of origin for that vehicle. If this is not available, the Washington state patrol shall assign a model year at the time of inspection." But it's not clear to me that the patrol would necessarily assign it the *current* year, i.e., 2005-- it seems reasonable that if the chassis and chassis VIN is from a 73 vehicle and the suspension and drivetrain are consistent with a vehicle from that year, they could reasonably choose to assign that year to it.
Chris Sutton (whose build diary is featured on this site- see "Building a Speedster Kit" on the home page) built and registered his speedster in the Seattle area and if he's still around may be able to add some valuable perspective to this discussion, as could anyone else on here with a WA speedster.
Good luck, and keep us posted!