I disagree with the answer above on one thing:
1. Mid-June in the High Sierra can be pretty darn snowy. Expect snow levels to be between 7,000 and 10,000 feet. And while this year is a lighter snow year so far, they are forecasting more storms to come this week. Since most of the John Muir Trail is over 7,000, and a lot times it goes well over 10,000 feet, you are going to see a lot of snow. And there is no way of knowing, right now, what the conditions will be. If we have a cooler than normal spring, as happened last year, you will still see a lot of that snow on the trail. Or over the trail.
If you've never used an ice axe or crampons, I would not suggest learning on this trail. Forrester Pass, at 13,000 feet, has a famous "chute" on the south side that is very steep and always packed with snow. And the north side is very snowy. If you have never hiked long distances or climbed steep passes in snow, I think you should consider an ice axe. As an example, most people don't hike the JMT in June. But Pacific Crest Trail hikers usually start the southern end of the JMT by mid-June if they are planning to finish in one year. And they ALL take ice axes.
2. Given that situation, do you still think you can hike this in 15 mile days? It's a lot harder to hike in snow than on a dry trail.
3. It sounds as if you are planning to do this one with a re-supply. Most people pick up a package at the mid-way point for this hike, but there is no guarantee that this will be possible in mid-June. Probable, but no guarantee.
4. Remember that the national parks require use of a bear canister. that will limit the volume of food you can take, and also add weight to you pack. Most people have a hard time getting two full weeks of food into a single bear canister. And the two end points, are both in national parks, so you'll have to use one right from the beginning.
Now for your questions;
* How crowded will it be during this time period?
Not so crowded, for obvious reasons, as noted above.
* Are bugs going to be a big problem and annoyance due to the snow melt?
No. They will be a big problem and annoyance about a week AFTER the snowmelt, and for another month after that. Last year, with the huge snowfall, it was a banner year for bugs.
* Will there be lots of snow on the passes? Will that require an ice axe for safety?
See my note above.
* If there is a good amount of snow, think I can still get away with doing the trail in some good trail runners, or will I need boots?
Lots of people do this trail, and the PCT, in trailrunners. But you may find yourself wishing you had something a bit better for steep icy snow pitches.
* Are there any trails that are about as long that can compare to the JMT for grand vistas and mountains?
Sure, you can hike the whole length of the Sierra without going on the John Muir Trail. And while some of it parallels the JMT, a portion of the Pacific Crest Trail is also an option, as it the Tahoe to Yosemite trail. The JMT is probably the most scenic of all (the PCT follows basically the same route through this area) but there are lots of other options.
And while it sounds like you are interested in an "epic" hike with a name, you could also hike for hundreds of miles in the Sierra far from people on other trails. Our website has something like 60-70 different hikes on it, and only about three involve the John Muir Trail.
* What temp rating down sleeping bag should I get? I tend to sleep warm. Will I be hot in a 20deg bag?
Nope. It can snow in June in the Sierra (and any other month, as well.) You can always open up your bag to cool off, but we use a 20 degree bag pretty much all summer at high elevations.
Here's our website; http://sites.google.com/site/backpackthesierra/home
Hope that helps