They didn't have valet when I was there last month. The self parking, in the basement, is thirty-something dollars a night. It wasn't free to hotel guests.
Parking is basement parking. I do not believe there was a charge for self parking and as long as you were staying as a guest. If you do Valet then there is a charge. Double check with hotel direct as there also may not be a charge if you have status with the chain.
We didn't park. The free shuttle goes from the hotel to waterfront every half hour and is walking distance tot he monorail and other public transport.
Parking is under the building. The cost is somewhere around $35/day I think.
We took the light rail to the hotel, and then Lyft back to the airport. If you call the hotel, they certainly will be able to inform you about parking options.
Get an answer from Hyatt House Seattle/Downtown staff and past guests.