GD in June-ish
Joe...can't help you on rendezvous locations in the south but can make some comments on heading north.
I rode the route in July 05 from Roosville to Salida. Everthing along the way was open - although some of the passes near Buena Vista were still snowed in (e.g Tincup, Tomichi) but they aren't really part of the GD route.
I got patchy snow only in the Union and Togwotee Pass areas east of the Tetons in Wyoming. Pinedale area is known for being cold.
You might get some snow in the Colorado passes but most should be open (e.g. Boreas near Brekenridge) .
Might also be high water in some of the creeks and plateau areas that the route goes through...usually it's a low lying area that muddies up with the melt....as you said, there's usually a work around.
Further north you may also get some patchy snow/water/mud on the Big Sheep Creek byway in southern Montana and maybe in the Stemple Pass areas where things climb a bit.
Lots of the route north of Helena through the Flathead area follows valleys where the snow should be down although passes may still have some snow in the shaded areas.
All in all though you should be pretty much OK although this will depend on the actual depth of this year's snow pack.
The cycling maps prepared for the route give good detail, but are limited for developing work arounds. In my trip I used Michael McCoys' book and the Delorme Gazetteers, which were pretty good in letting me see options and alternatives. I put them in a clear map case and strapped them to the top of the duffle for ready access.
Good luck in your planning....it's a great route....
Here's a A few GD Pics
that may be of interest.