what do you mean by "richness"? Do you mean someone who has more than ___ in assets? Or no assets but makes more than ___ a year? Or someone who lives below their means and always has money when the kids need something (eg someone who doesn't splurge on them self but the kids are never left wanting)? Or someone who lives beyond their means and convinces them self they're successful while they keep adding more and more to their credit card debt?
It also seems like you're asking if how people treat the "rich" person affect their happiness. What if the "rich" person never tells his peers he is rich? People can only treat someone differently if the person makes a point of letting everyone know. Perhaps a kid who boasts about his weath is treated differently not because of the money but because he's a braggart?
Personally I've spent time with wealthy people (as in assets) and high income people. If they're unhappy, it has more to do with them. It's someone else's fault if they aren't happy. The one very wealthy family I know seems miserable. The kid is spoiled rotten and is now in her 20s and had to "accidental" kids by 2 different guys she doesn't even like. The father is jaded and he spends his time buying more and more because a few weeks after each new purchase, he's bored again. The mother seems lost and despite having what most consider a fortune, she can't stand up to her pushy husband or give herself permission to enjoy things. The two "accidental" kids are bought every toy there is by the entire family, spoiled with too many gifts, but nobody wants to spend time with them. Money does not buy happiness.
That being said, I've got some relatives who are worth millions. They're down to earth people, and you never know they're well off. They have good days and bad, just like everyone else. They're happy, caring, giving people.
You ask: should anyone who is well of be expected to give back? I think most people give, even if they're not rich. Middle and lower classes just give smaller monetary gifts or may do volunteer work. I personally am still unemployed, after being laid off in January, but I can do charity work. I don't believe anyone should feel it's a mandate or obligation though.