Hi A&W. Welcome to the ADKs. You asked about the time period between Memorial Day and the 4th of July and that's pretty close to the black fly season in the Adirondacks. There was a thread just recently where people discussed black flies. It's in the Q&A forum and the last posting was just a few days ago.
As far as other conditions, generally, during that time of year, you might still get some muddy spots from the Spring thaw, but the amount of mud would depend on weather conditions too. There's quite a bit of variance in mud, bugs, etc depending on the elevation, sun and wind exposure, and location. Right now, we're starting with a very dry Spring and if it stays this way, that would affect not just the mud, but the bugs too. Another thing I always consider is the temperature on the peaks. Even when it seems nice and warm at the lower elevations, I always bring enough extra layers with me to prepare for freezing temperatures at the summit. Even if the temperatures aren't freezing, the wind chill factor can have a big impact.