I would recommend you to hire a 4x4, "just in case".
Few others would make that recommendation based on the region of exploration.
Can there be snow in June
Not where you are going. But strong winds and heavy rain can hold you up a bit and certainly affect your sightseeing. The sightseeing times on tripcreator seemed short and you had lots of rather tedious things of marginal interest, almost like you were trying to see everything there was without regard for interest level, and would end up rather regimented, come on Candace-Marie, just 40 minutes for this stop, and that includes the walk to and from the car and your visit to the toilet. (That's intended to be a reference to a TV drama "Nuts in May" by Mike Leigh.) The walk to and from the car park is sometimes rather long in Iceland, especially for natural attractions. Things like "Snaefellsjokull NP 3 hrs" are ridiculous. It's a mountain, not a pleasure garden, how can they say what you intend to do there and how long it will take.
there seem to be no mud roads on my trip
The road east of Stykkisholmur isn't paved all the way, so probably about 40km, and it isn't 4wd road, but it is bumpy.
My wife is not going to drive, so I decided to plan a somewhat relaxed trip - 1249 km in total, and just 3 different hotels.
I can't see what you are now proposing but personally I have wouldn't bothered with staying near the airport on your original itinerary. I wouldn't call 1250km in a week particularly relaxing myself.