We always use Blue Car Rental for Iceland and will again in September. They are local, professional, and we have always had good experiences with them. They have contactless pickup and drop-off which saves a lot of time as well. Their location is literally across a small parking lot, so a 2 minute walk. You can see it from the terminal. No shuttle needed.
As for where to go etc, is this stop on your way to Europe or the way back? If this is on the way there, pick up the car the morning of arrival and head into Reykjavik and stay there for the night. We too fly from SEA and know how hard jetlag can hit, especially when you are behind the wheel. Play it safe and stay nearby and enjoy the city for that first night, and then head out in the morning. Assuming you have 2 other full days (?), I would head up to Snaefellsness the next day. It is a long day, but so many sights and it is sort of like a mini Iceland in that you can see so much in a smallish area. The next day I would do the Golden Circle and a bit beyond. Do plan to stay no further out than Selfoss on that last night in case of bad roads/weather so that you do not miss your flight the next day. Another option would be to take a 2 day south coast tour and leave the driving to the tour operator. I really like troll.is for these types of trips as we have had good luck with them.
Places we have had great stays in Reykjavik include 4th Floor Hotel, Hotel Fron, Center Hotel Laugarvegur, and Center Hotel Midgardur. The last 3 of these all have free parking on site.
In Stikkisholmur, up on Snaefellsness, I love Akkeri Guesthouse. This upcoming trip we are staying at the Elligsen though as the other guesthouse was full for our dates.
In the Golden Circle, take a look at the Blue View Cabins. If the northern lights are showing, this is an ideal location to stay. We loved our stay there even in the summer though;) On our upcoming trip we are staying at the Arbakki Farmhouse (Reykholt) as well as the Hotel Eldhestar (Selfoss).