I’m not normally an over-insurance type, but the rental car sand and ash cover is probably worthwhile if going round the south coast and across the outwash plains having experienced that type of weather... it can sandblast the paint right off. Also watch the car doors when it’s windy... we saw one get bent right round and last time I was there on a field trip, a panel of the coach blew off when parked and disappeared into the sea.
Yes, all your suggested route is on proper roads. Don’t take a Kia Rio on an F road unless you want punctures and a flooded engine. They are washboard gravel, covered in rocks and have deep rocky fords.
You really don’t need to join a day trip - it’s all very easy to find and there is 4G everywhere. Just stick to the tarmac roads.
If you do one thing - walk up to the Reykjadalur hot springs at Hveragerdi - once parked next to the cafe it’s about an hour walk up and a bit less back. All on made trails so very easy going, if a bit uphill initially. It’s a hot/warm river half way up a volcano with bubbles coming through the bed. It’s one of the most incredible places to sit in a river and admire the view - you must go. It’s not far from Reykjavík. You can change into trunks etc once there - wooden fencing provides modesty.
If you’re going round the south coast, on the way to/from Vik make sure you visit Seljalandsfoss and Skogafoss waterfalls as well as Solheimajokull glacier (these days you can park pretty much next to it). As you look at Seljalandsfoss from the car park, there is a hidden waterfall in a canyon about 750m to the left. It’s signed (I think) and is unmissable. At Skogafoss, make sure you walk up alongisde the waterfall and then keep going up the path - after about a mile you get to a series of incredible views into the gorge. Also the black sand beach and columnar rocks at Reynishverfi (or something like that... it’s right near Vik).
For the true Icelandic egg experience, stop
at Krysuvik on the way back to the airport. It absolutely stinks but is beautiful and unique.