What Moder-dye said.
Also there's an 11 year cycle of solar flare intensity, the last maximum was in 2013 (cycle 24) and will be at its lowest just about now (5.5 years in).
Iceland isn't exactly a good place for 'the' lights as it's relatively far south.
Lofoten, Finnmark etc. are much better. More regularly, more intense. That said, once September - my favourite time to travel in Iceland - has arrived chances to see the lights during a multi week trip are very high (some exceptions can be encountered obviously...)
I remember one winter trip to Iceland where even the road between Reykjavik and Selfoss was closed for about 24 h due to large snow drifts.
This is rather rare, but you'll need to be flexible.
If it's for Iceland and the lights only, go in Autumn, if winter has a big place in your plans, go for it.
Gear choice sounds reasonable. The links you found are your best shot to find out about the road conditions.
*goes and looks for a link to a handy OSM based map of Iceland used on all trips*
During a double traverse of the interior in 2016. Trip of a total of 3 weeks, ending mid September.
I had 3 nights with lights during that trip. During a 3.5 weeks trip, mostly hiking across Iceland during the same time of year in 2013 I saw nil. The weather wasn't quite as good in 2013 as it was in 2016 though.
Also check for AirBnBs. In 2016 I arrived in Akureyri with most Hotels/Hostels fully booked but was able to get a lovely room through AirBnB.
Thinking about busses... they can come really handy, also for cyclists. It's rather convenient how the bus companies handle the needs of cyclists, but I don't know if they run the ring road in winter.
One of a few operators:
https://www.nat.is/travelguideeng/bus_stop_to_hofn.htm
Doesn't seem to run in winter, but worth asking.
There's also Sterna and Reykjavik Excursions I can think of.