Surprisingly enough, I agree with Gwazi too. o.O
(Big rant on the way. Go to the last 3 paragraphs if you just want a straight out answer)
May I vs could I is really just nitpicking, and it's more a generation thing than anything else. It doesn't mean that people are loosing their manners, it just means that times are changing and how we're using our language is changing. I'm not a native English speaker, but it's not like it's much different from what's happening with my own language. The way I spoke as a child was completely different from how my parents spoke as children. I was raised with good manners just like my father was, but still our speech was different since I came from a different generation and the language (just as society) had changed slightly since my parents childhood. It's natural changes and we can't expect the next generation to talk exactly as we did as kids.
I do think children should say please if they ask for something from their parents, and thank you when they get something, be it compliments, objects or a nice gesture, but when they are with friends, dropping their manners isn't a big problem. We speak differently when we are with different people, and it's important to not force a kid to always keep up such an act if it's not part of their personality.
I don't particularly see a drop in good manners, there's only a slight difference in the way people speak from when I was a kid. Some children are loud and obnoxious, but most of them aren't. Two years ago, I worked as a temporary teacher and day care person. When I took care of the 5-6 years old kids, I found ONE ill mannered student in a group of over 30 students. When I worked with 8-11 year olds I can remember 3 in a group of 60 (They were technically 2 groups, but they shared the same space and were only divided during snack time).
When it came to the older kids that I actually had classes with it was only the 9th graders that were ill mannered, loud and un-respectful towards me as a teacher. The 6th graders were pretty much a class filled with angels, and the other classes were very silent and respectful too with maybe one or two that had to be reminded not to disturb the class.
It was kind of a surprise since it's when a temporary teacher shows up that they often are out of control. (Experience from my own childhood.) So if anything I would say it has gotten better since I was a kid in the 90s and early 2000.
It definitely does depend on where you live though. It changes from country to country and city to city, and from house to house. Different families raises their children differently, different schools takes care of troublesome students differently, different neighborhoods have different degrees of manners, thus there will be a difference wherever you go.
So to answer your questions:
Yes I think that a parent should teach their kids to be polite, but we shouldn't expect them to be polite all the time. Most people wouldn't be polite around their best friends. When you're extremely comfortable with someone you don't really care to say please when asking to borrow a book. (For some it might feel natural, but for others it won't). Teach them to be polite towards parents, authorities and when they're in public, but don't expect them to be polite all the time. If a kid is tired they will act differently and maybe be more annoyed and maybe talk less. They're kids, we can't expect them to act like adults 100% of the time.
Asking someone how they are is unnecessary if you don't actually mean it. The only time I ask is if I know someone has had a rough time, and I usually don't just ask "how are you" but add "I heard about this or that" so that they can be honest with me. How are you is one of the most dishonest ways of showing that you care. I don't care if it's good manners or not, no one will ever change the answer to their real feelings because they know the question isn't real. It's just like bringing up the weather. It's a way to get the conversation going without caring what you're talking about.
Can I have that or may I have that is just nitpicking and doesn't matter. Teach them to add a please instead of complaining about a can or may. Can I please have that? That's polite enough in my book.