If you are coming in May, then definitely self drive. Staying the first night in Reykjavik is smart. We also fly out of SEA and have been to Iceland a number of times and know how brutal the arrival day can be. Spending it in Reykjavik is a great way to get over jet lag and see what all the city has to offer, which is quite a bit! There are the local pools, the Citywalk "free" walking tour (tip at the end), the food tour, Harpa inside and out, the Sun Voyager, the wharf, rainbow road, go to the top of the church for amazing views of the city, Braud for cinnamon rolls, etc. Plus a variety of museums. We. use Wakeup Reykjavik for the food tour and they have a coupon right now. They use the original food tour. We are also doing that again on our Feb. trip.
The drive up to Jokulsarlon is a good 5 hours each way, just driving. So the next morning head up that way, but plan to stay the night up there and then on your way back the next day you can easily fit in the golden circle. No way would I drive there and back in a day as there is just way too much to see.
If you want to do a lagoon, either do that upon arrival or if you have time on your last day before your flight. There are a number of options. TheBlue Lagoon is wonderful and perfect right when you land or right before you go to the airport. Sky Lagoon is in Reykjavik, but on the outskirts, so you would want to drive. In the Golden Circle there is Fontana and Secret Lagoon. Just north of Reykjavik is Hvammsvik. We have done Blue and Sky twice, and the others once. Sky has the 7 step ritual which is sooo good and we are actually headed back to it for the 3rd time soon. Hvammsvik is hands down our favorite commercial one in all of our trips to Iceland.