1. "...but what exactly does this accomplish"
Nothing I have found in almost 30 years of playing.
2. "I notice that I'm almost always on the sharp side when I look at my tuner"
Virtually all diatonics are tuned slightly sharp to begin with because so many beginners unintentionally bend the notes and make them flat (out-of-tune). Standard tuning is considered A440hz, but most diatonics come at about 442 or 443hz. (Also see answer number 5 below).
3. "Are new reeds harder to bend on than an older, well used set?"
Yes, slightly, but with good bending technique and a good quality non-leaking diatonic to begin with, it is a minimal difference.
4. "Will they last longer with a proper break-in?"
No. I do not know a single working pro that breaks in their diatonics. As far as I'm concerned, diatonic harmonicas will never better than when they are brand new (although the first thing I do to a new harmonica is tune it to where I like it and increase the gapping where needed).
5. "And what exactly is considered playing gently?"
That is a silly notion that is still being printed up with new Hohner harmonicas (mostly because the harmonica companies get so many harmonicas sent back due to poor beginner playing technique -- hole 2 draw is the number one complaint, and it is almost never the harmonica).
It does apply somewhat to chromatics when being played by beginners, but for the most part, one needs to learn to breathe correctly in order to do the least amount of damage over time to the harmonica. The correct angle of airflow will either mean bending or not bending. Constantly playing with a slight angle of airflow will make your playing consistently slightly out-of-tune (flat) and doomed to a shorter harmonica life span.
6. "Is trying to learn to bend on them being gentle?"
Don't worry about it. As you improve your harmonicas will last longer than the first ones you got. You've got to break a few eggs when making an omelette.
A lot of the junk out there you hear about harmonicas and harmonica playing technique is based on 50 or 60 year old information that keeps getting passed along. Much of the older Hohner literature is guilty of this as well as many of the cheaper instruction books and materials available.