We have had good luck with the northern lights in Iceland. On both of our October trips and our February one, the lights were out. For the October trips, they were full on, even right in Reykjavik itself, and the viivid greens were easily seen with the naked eye. Yes, that is not the norm, but it does happen!
We always self drive in Iceland, but one thing you really have to keep in mind with that is to keep an eye on both the Vedur and the Safetravel apps multiple times a day. This is for road conditions and weather, as it changes literally every 5 minutes there.
Some activities you absolutely have to have a guide for, such an ice caves and glacier hikes, so if you want to do those then I would suggest skipping the car and doing a 2 day tour up along the south coast where they incorporate these activities. We have used Troll for these and they were great!
For the other days, you could wait and see how the weather is and just rent a car from downtown Reykjavik. Blue has an office there and that is who we always rent through. We did that on our February trip as that week was back to back storms, but one day was beautiful so we grabbed a car and off we went.
As for the NL, it all just depends of course. The apps are ok, but honestly IME it is just looking up. Sometimes you will see what you think is a cloud, but watch it, and see what it does. We did take a NL tour once, in late October, and we used Aurora Experts. Heimer is excellent and I would go with him again!
Other day trips, whether self driving or a tour, would be Snaefellsness, the Golden Circle, and the Reykjanes Peninsula. If you do Snaefellsness, do not miss Hvammsvik Hot Spring which is about 45 minutes north of Reykjavik. It is our favorite commercial hot spring in all of our trips. The GC you can also easily self drive, or do a tour, but try to see Bruarfoss and Fridheimer (the tomato farm/restaurant). Bruarfoss has a new parking lot so you can just walk there in a minute, or you can do the hike which entails 3 falls and is really worth it. Fridheimer you need reservations, but it is very good and you eat inside the greenhouse with the plants all around, and bees, but don't worry the bees don't hang out in the dining area. They have horses there as well. The Reykjanes Peninsula is an area that many visitors do not explore. It has a lot of fascinating things, many of which you will find just by driving and not on a tour.
As for restaurants, think about doing a food tour. We head back in a couple weeks and will be doing it again as my daughter will be with me and she has not done it. Book through Wakeup Reykjavik for the original tour. It is 3 hours long, but so much food (all very good!) and will cover you for both lunch and dinner. Otherwise Messinn is our favorite restaurant in Reykjavik and we return on every trip. Forrettabarinn is really good, small plates and a bit off the tourist path so when we have been, we were the only American's in there. Honestly, the only place in all of Iceland so far for us that we did not like was Tapas Barinn in Reykjavik. Crammed in, food was fine, but the tables were so tiny that they barely held two tapas plates and two drinks, and there was no room between the tables. They should have just changed it to big communal tables. Just overall, it was not a place we enjoyed, FWIW. If you make it to Selfoss on a self driving tour, there is another Messinn location there as well, and just outside of town is Ingolfsskali Viking Restaurant that is excellent.
If you enjoy lagoons/hot springs, Blue is the obvious one, and it is good, but there is also Sky just on the outskirts of Reykjavik and I like this one a tad better than Blue. Their 7 step ritual is heavenly. We are going there for our 3rd time on our upcoming trip.