Seattle to Butte Yellowstone Rapid City Medora.Best Route?

Yes
Yellowstone is "open" year around. It's just not all the roads are "open". If the roads are closed the buildings may also be closed, and so are the campgrounds.

The only road open (to vehicles) is the north entrance at Mammoth.

When the rest of the roads open depends on the year, a "low snow" year it could be April. A heavy snow year, well, I've had a snow-ball fight (from fresh fallen snow) on July 4th. I've seen years when the passes aren't opened until mid June.

On average it snows (a little) every June.

It really depends on the year, most El Nino weather patterns lead to a low snow pack (in 1977 I drove through the entire park in early April). So this -might- be a low snow year. But maybe not....

I used to live in the area and have been in the "park" every month of the year.

Yellowstone National Park is open year round. Most, if not all, of the park's campgrounds don't open until May. Check the parks web site for the right information. Most likely what you will encounter in the park in April is snow, snow, snow. Even if you stay in West Yellowstone you might not be able to get into the park from there due to weather, spelled snow. Friends of ours went in early May. Fishing Bridge camp sites had been plowed, but not the area between them so snow was piled real high.

Yellowstone does not open until mid May.