Iceland is a place we go often, and we just returned from yet another trip:). You are correct that they do not have time for the ring road. Here are my thoughts.
Day one is perfect for arrival day. A food tour and/or the Citywalk "free" walking tour are excellent and are great ways to get an introduction to the city/country. We have done both several times. The food tour may appear to be expensive, but there is so much food that it easily covers lunch and dinner, so in that respect it makes it more economical, plus it is just a lot of fun. The walking tour books about a week in advance and it sells out quickly, so if they are interested, mark the calendar to book it.
Day 2/3 with Troll is good if they want to do an excursion such as Katla or a glacier hike. Otherwise I would just have them rent a car and explore on their own. If they do, I highly recommend exploring Gjain. It is gorgeous this time of year, and there are very few people.
Day 4 I would not bother with hanging around the city. They could easily do the Golden Circle this day. If they rent a car on day 2, they could do the GC on their way back down, and still be back in the city in time for a chance at an aurora. BTW, Aurora Experts is who we have used a few times and he is wonderful! We were booked last month, but both times the weather was not conducive, unfortunately.
The GC only needs one day. With the extra days, have them ahead up to Snaefellsness for at least a night. There is so much to see up there and it is like a condensed version of Iceland. We personally like to stay in Stykkisholmur. Take a look at either the Akkeri Guesthouse or the Hotel Egilsen. We have stayed at both, and the time I stayed at the Egilsen was with a girl friend, FWIW.
If their flight is early on day 8, they will want to stay near KEF as you often really do need the full 3 hours there, and Reykjavik is 45 minutes away. So the Aurora Hotel, which is a short walk across the parking lot to the terminal, is perfect for very early flights out. If the flight is a big later, say late morning, then I highly recommend the Northern Light Hotel in Grindavik. That is about 20 minutes away and basically right next door to the Blue Lagoon, which they could do that last night.
We do love camping in Iceland, but I would not recommend it for them for this trip. Mid to late September is when the weather can start to turn and the winds can and do blow camper vans off the road. Also, most campgrounds close by mid September, so they would need to plan ahead and see what is actually open.
The lagoons are wonderful! We always try to hit as many as we can on our trips. The pricing can very by quite a bit, but honestly I would not nix anything due to price since each one has its special qualities and IMO they are worth the price. The BL is the one most people think of , and we do like it. Sky lagoon is right in Reykjavik, although not walkable and the bus requires a transfer and then a long walk. So take either a bus transfer through Reykjavik Excursions or a taxi. A taxi is about $30. In the GC there is Secret Lagoon which is the cheapest. Then Fontana is there too, and they are currently undergoing a renovation. We loved it ,but I am sure it will be even better when it reopens. Then there is the brand new Laugaras, which is now tied as a favorite with Hvammsvik Hot Springs for us. Hvammsvik is just north of Reykjavik about 45 minutes, so is perfect for on the way to/from Snaefellsness. This one sits right on the fjord, so a great cold plunge. Also on the way to/from Snaefellsness is Krauma, which we have not made it to yet, but have heard good things about. As well as these lagoons, take a look at local pools which are in each town and are only about $8 US dollars or so. Sundhollin in Reykjavik is the one we always go to there, and they have rooftop hot tubs. In the GC there is also Hrunalaug, which is tiny and rustic, but we loved it.
