The only thing that happens when you're rich is that you can help more people. If you were not helping people before (whatever that means to you), you probably won't do it after.
What you said intuitively appears true, but it's actually not the case in real life. Helping others is not limited by time or money - you can help others in large or small ways. And the OP is absolutely correct, if you are not already helping others in some way, having more money won't change your behaviour.
On a related note, I was talking to an university alumni director in Australia (where philanthropy is not culturally ingrained like in the US) and she was telling me the statistics on donations. If a student did not make a donation to the university within 5 years of leaving, 96% of them would never make a donation. However, if they made a donation within 5 years, a very large percentage of them would continue to make donations (regular or irregular) over their lifetime. How much they initially donated was irrelevant - it was the initial act of donating that determined the probability of them continuing to donate.
Similarly, if you are not prepared to donate a few dollars when on a 'low' income, you are unlikely to change your behaviour when you are on a 'high' income.
Agree with parent, disagree with grandparent. If there's one thing you learn when you're doing philanthropy, it's that people who give to charity this year will probably give to charity next year; and people who are planning to give to charity "next year" will, next year, be planning to give to charity "next year". Donors donate, nondonors don't.
Not really. If you are rich you may have some investments you would like to make and, in doing so, you are helping other people: you create jobs, you generate wealth for everyone, you might even make the world a better place depending on what your investment is.