1. For the collectibles, do the bugs ONLY appear where the briar patch was, and do the rocks ONLY appear where the mosaic is?
Yes and yes. And shells only appear near the shore.
2. What difference does it make if you set the game on easy, normal, or hard?
Honestly not sure; I never played easy more than once, and that was only in VV1.
3. When I try to train a master parent, I usually try to choose a girl, and I have found that having them 'go indoors' is the only way to improve this skill. But when you drop an adult on a child and they give lessons or tell a story, it sometimes says 'This villager has improved at parenting!' But I find that no change in their skill level has occurred... What is up with this? Because when they 'go indoors' and make a baby, the mother can't work on improving that skill for the entire time she is 'caring for her baby' so I'm a little perplexed.
The bar does not increase smoothly, it increases in increments. While your first success in a skill is enough to earn the first increment, as your villager gains skill, more successes are needed to earn each increment.
Telling stories or teaching children DOES earn parenting skill. You'll notice that highly skilled parents, when they 'go indoors', also often have no noticable increase in skill after a success, for the same reason you often see no increase after telling stories, or fishing or fixing huts, for that matter.
4. How do the parents pass a skill down to their child? I find that even when I have my Esteemed Elders father children, no skill is passed to the child. It seems to be random. Also, is it possible for a child to inherit more than one skill? And more than one bar of a skill?
Actually, I'm pretty sure it is random, in VV2. In VV3 the genetics include passing on skills, but I don't believe that was implemented in VV2.
I haven't played VV2 in a while, but I don't remember anybody having more than only one or two increments on their skill bar.