I wrote the same info in a different thread, but I'd say a big yes to the ice axe (and training/practice) and a no to crampons. The ice axe is to PREVENT you from sliding in the first place, through proper technique and self-belay. As my instructor in my winter mountaineering course said (i paraphrase): "once you start sliding, you've already lost." You DEFINITELY don't want to think that that axe is going to stop you before you hit some rocks (e.g. Forester). So get one, learn to use it, and practice. I have an old Cassin Ghost, but if I bought one now, I'd probably just get a Camp Corsa 70cm one at REI. Also, remember that you look like a bad-ass in your photos at Forester if you are holding your ice axe aloft. :)
In 2004, I'd say you didn't need an axe, but that was a low snow year followed by a spring of incredibly hot temperatures, so the snow was greatly reduced when we hit the High Sierra starting June 10 or so. I'd say get it in Kennedy Meadows drop box, and send it home as soon as you don't need it.
Crampons, I'd say no, for a few reasons. In June in the High Sierra, the snow is probably only hard first thing in the morning. So it's easy to hit a pass a day NOT at that time. We didn't worry about trying to do two, and in general we slowed down to 18 or so miles a day in the High Sierra... it's a bit more tiring hiking, and it's so pretty, there's really no need to rush.
I took crampons when I did the Sierra High Route in 2006 in LOTS of snow. We were on snow for most of our hiking time. We did use them, but only to descend hard snowy passes first thing in the morning (passes we had camped atop the night before, which we did fairly regularly). I was happy to have them, but we could've timed things different or just waited a couple hours. I bet we used them for six hours on the whole trip. That said, cruising across hard huge suncups with crampons is WAY better than crossing huge soft suncups without them!
In general, I've come around to thinking that snow is my FRIEND in the High Sierra. I'll take it vs. loose steep talus any day.
heading towards Forester Pass in a low snow year (2004)