If you haven't used an ice axe before then you probably don't need one because if you had it and really needed to use it you wouldn't know how. At least that's my thought on the matter. It is safer to turn back than proceed without proper skill and experience.
It's a case by case basis right now, some mountains are in summer mode, some are not. I know Handies from certain routes is clear, same for Redcloud and Sunshine while Uncompahgre is still quite snowy in places.
This report from earlier in June may be of help: http://14ers.com/phpBB3/viewtopic.php?f=2&t=30363#p359661
I'll have a fully updated report from someone I know, hopefully by this time next week.
If you have a pack, shoes, trekking poles, clothing, sleeping bag, food-those items should be hung or secured somehow off the ground if they are not going in tent/tarptent. But again, not all campsites are created equal, some will be safer than others.