I think your itinerary looks pretty good. We have always had rental cars except our first trip as it was just a 3 night stopover and we had no idea what we were doing;) If you are up for driving, I do recommend that as it it gives you much more flexibility. I will try and gives a few suggestions whether it be self drive or tours.
For day 1, Reykjavik is perfect and there is so much to see and do. I love the Citywalk "free" walking tour (sign up about one week in advance, and it does sell out) and you tip what you want at the end. We have done that twice on arrival days and it is really good. Also, the food tour is excellent. We went at 11am and that was perfect as it goes for 3 hours and there is so much food that it covered us for both lunch and dinner. So in that respect it was cost effective as well as just a lot of fun. You could go to Sky Lagoon which is wonderful or just one of the local pools right in town. SL is in Reykjavik, but about a 15 minute drive. We just walked to Sundhollin as it was about 2 blocks fro our hotel.
On day 2 I would try to do the northern lights tour this night. Then if the tour doesn't go, you still have time to have a 2nd chance as most companies offer that. I highly recommend Aurora Experts. We went with him this past October and he was wonderful! Plus he is the only operator that has winter jumpsuits in case you get cold, and he doesn't charge extra for tripods etc like others. Whether you do a tour for the Golden Circle or self drive, do not miss Fridheimer, the tomato place. It is a wonderful stop for lunch. If you want to soak, Secret Lagoon or Laugarvatn Fontana are both in this area. We are headed to Fontana this summer, but have been to Secret Lagoon on a prior trip and loved that, FWIW.
For day 3 and 4 you could self drive or do a 2 day south coast tour. We have done both, sort of. We have done it ourselves, which I prefer as you can stop at any time. The tour we were booked on was in early February, so a very stormy time of year. Weather was turning and a storm was rolling in, so they had to cancel the 2nd day of the tour to stay safe. We used Troll, and they were excellent with communication and they refunded that second day immediately. We did the one day with them as it included a glacier hike so we had to have a guide. We also used them for an ice cave tour, as again a guide is required for that. If you are not wanting to do a glacier hiker or ice cave, then definitely self drive if you are up for it. If you are wanting to do either of these excursions, then it makes the most sense to do a 2 day tour that includes this. I highly recommend Troll as they were so good to work with, especially when things did not go as planned.
If you are self driving, on day 5 I would wander the Reykjanes Peninsula. It is basically the area where the airport is and along the coast in that area. I love this area as there are so few visitors that check it out, and there is a lot to explore. It would be a relaxing day and then you could use this night for your second chance at the Northern Lights.
Day 6 for Snaefellsness would be enjoyable.
Depending on what time your flight on day 7 leaves, you might be able to do the Blue Lagoon that day before heading to the airport. We had a really nice stay at Northern Light Inn, literally about a 2 minute drive from the BL. We stayed here on our last night on one trip and it set us up perfectly for the BL in the morning then to the airport to return the car and fly home. If you do the Reykjanes Peninsula, then it would fit in perfectly with that.