Thanks to Daniel and Rajan for very useful information. I have a pretty close to final itinerary now, but not sure where to stay for each night. In parentheses when knowing where to stay:
Day 0 - (Reykjavik, June 16)
Day 1- The Golden Circle
Day 2 - Selialandsfoss & Skogafoss
Day 3 and 4 - Vik coast
Day 5 - Moss covered lava rocks & Fjaorargljufur Canyon
Day 6 and 7 - Jokulsarlon Lagoon, Ice Beach and Skaftafell National Park
Day 8 - Vesturhorn mountain and Laekjavik coast (Hofn, June 24)
Day 9 and 10 - Landmannalaugar (Hrauneyjar highland, June 25-26)
Day 11 - (Reykjavik June 27)
Day 12 - Fly out June 28
Questions:
1) Any suggestions on towns to stay in for Days 1-7? I am awaiting a map I ordered that may help me also.
2) Will I be able to get to other places in the interior highlands on June 25 with 4WD vehicle, or is Landmannalaugar the limit for that time of year? If so, should I allow more days Hrauneyjar highland and less elsewhere?
Any other suggestions welcome.
Quickly -
It is not necessary to keep changing hotels. For example, Seljalandsfoss, Skógafoss, Vík can use one common base. I recommend Hótel Fljótshlíð near Hvolsvöllur (you will have a beautiful view of Eyjafjallajökull; this area is also of significance in Icelandic sagas). Second choice - Country Hótel Anna further up in the Eyjafjöll region. All these hotels have Tripadvisor reviews.
Fjaðrárgljúfur and Eldhraun ("Fire lava," which is the moss-covered lava) are just outside Kirkjubæjarklaustur. I recommend Hótel Klaustur. From here you could also take the Nyrðra Fjallabak (F208) route all the way to Landmannalaugar. The other approach to Landmannalaugar is from Hrauneyjar (which you can get to from Hella or Hvolsvöllur). Don't be fooled by the "4WD" label if you plan to take the F routes yourself. Most of the 4WDs rented are city SUVs and although car rental companies will tell you they are suitable for F routes, you are ultimately responsible if things go wrong. There are glacial rivers to ford and all insurance becomes invalid if you incur damage while crossing rivers. Rent a high-clearance robust vehicle such as the Toyota Landcruiser and take EVERY kind of insurance offered (if peace of mind means anything to you).
For Jökulsárlón, I recommend Hali Country Hótel. (Avoid Guesthouse Gerði, which is its neighbor unless you have absolutely no choice.) The Skaftafell area is an easy drive from either Kirkjubæjarklaustur or from the Jökulsárlón area, so you don't have to take accommodation there. But if you want to, there's Fosshótel Skaftafell which is quite nice. There are also two new Fosshótels now - Fosshótel Jökulsárlón (not close to, but within easy driving distance of, the glacial lagoon) and Fosshótel Núpar (alternative to staying in Kirkjubæjarklaustur).
Finally - give your planned itinerary a second thought. One possibility is, after the Golden Circle, drive all the way to Höfn and start your south coast exploration driving westward. On your eastward drive to Höfn you will be able to take mental notes on what you may want to come back to later for a more detailed look. Also, you will have two possibilities of getting to Landmannalaugar (stated above). Your final night (Day 11) can be spent in the Hvolsvöllur area instead of Reykjavík, especially if your Icelandair flight back to the US is around 5 pm in the afternoon.