You can remind your family that Iceland is rather safer for cycling and camping than Canada because there are no bears, mountain lions, wolverines or snakes, very few biting insects, one of the lowest crime rates on the planet, and, with the application of some common sense, the water from the streams is safe to drink.
You will meet plenty of other cyclists, including unchaperoned females. You will have no trouble camping in the campgrounds. And you will have ample opportunities for wild camping, on the ring road mainly in the remoter parts of the east, but otherwise more likely if you leave the ring road.
I've written some extensive briefing pages (in two parts) on cycling in Iceland plus some travelogue descriptions accessible via this page. http://www.masterlyinactivity.com/ivan/index.html
Personally I would not recommend sticking exactly to the ring road in the west of Iceland, or indeed in general, because there are nicer places to cyclein much of the country. Only in the south-east and the eastfjords is there no alternative, and fortunately this is one of the best bits of the ring road to cycle. But there are some bits actively to avoid:
Heading north from Reykjavik:
*The Akranes tunnels is closed to cyclists
*From Reykjavik to Borgarnes the road is very busy, often narrow and without shoulder
*From Borgarnes to Blonduos the road is quite busy, often narrow, without shoulder, and the scenery is the least interesting on the ring road
*From Blonduos to Akureyri the road is still quite busy, often narrow, without shoulder
Heading east from Reykjavik:
*From Reykjavik to Selfoss the road is very busy, often narrow and without shoulder (there is a plan to widen it)
*From Selfoss to Hvolsvollur the road is quite busy, often narrow, without shoulder, and the scenery is not so interesting
Assuming you are happy with untarmacked roads, I would strongly recommend you consider avoiding the western part of the ring road by following the Kjolur route (described in my notes). You can get there via Thingvellir, Geysir and Gullfoss, which most people want to see, as they are some of the leading wonders of the country. You will need to pack up about 5 days food from Reykjavik because on that route you won't really go past anywhere to buy some until you get to the north of Iceland. An alternative possibility, if you have the time, is to visit Snaefellsnes and the Westfjords.
I would also strongly recommend you travel via Landmannalaugar in the south of Iceland, which will cut off the ring road from just west of Kirkjubaejarklaustur to near Hella, using the "most popular" route as described in my notes. For many people, this is the highlight of their cycling trip to Iceland.
The only note of caution I would apply is that you do need stick-to-it-ness to cycle in Iceland, because some people can find the weather conditions more trying that they are prepared to put up with, especially the wind. But from your description of yourself, I think you probably have the confidence and persistence to make it. In theory, the winds should be slightly more favourable if you go clockwise, but don't blame me if it doesn't work out like that for you.
