Hey mari, I'll be on the opposite end of the trail this year (section hike from VT to ME), so I probably won't see you out there. As for your questions...(1) June is a tad late...the most popular starting dates from Springer are late March and early to mid April. I found that there were definitely hikers that started much later, however, they moved a much faster pace right from the start. You might find yourself a bit behind the main push if you start in June, but nothing to be too worried about. (2) As long as you respect a bear, and exercise your common sense, then you should be fine...keep your distance, especially if you see cubs. There are some great threads on this forum about bear safety. Some hikers might disagree, but I'm a firm believer in always hanging your bear bag. I find that most of the bear horror-stories that I heard were a result of a hungry bear sniffing out a food bag that was within too-close proximities to hikers.
Good luck with your adventure! The southern portion of the trail was so magical