Just a few basic thoughts.
As others mentioned, it's too early in the season to know what the snow pack will look like in June but I would say that there's usually a lot of snow in the areas you mentioned. Early June in the Snowy Range will most definitely be a snowy proposition. Hwy 130 which crosses the Snowies is usually opened by Memorial Day weekend. If the road to the TH near Sugarloaf is still snowed in, which is not unheard of, you'll need to park along Hwy 130 and walk in. As far as I know, Elk Mountain is entirely on private property.

There's a high probability that you'll need snowshoes in the Snowies and the Gros Ventres to reach your peaks.

As far as trail runners, I seriously doubt that they'll be of any use except for your 6,000 foot hiking, at least in June, in Wyoming. Are you talking about Wyoming when you say Most of the time, we will be hiking along at about 6,000 feet in warmer weather.??