The Royal Family have public money and private money. Their public money is very carefully documented, and we taxpayers pay about 63p each a year towards them. (I think this is just under a Euro and just over a US Dollar.) I for one don't get bitter and twisted about this. I think it is very good value for what we get in pomp and circumstance and all the general an charitable duties that The Queen, Prince Philip, Prince Charles, Princess Anne, Prince Andrew, Prince Edward, The Duke and Duchess of Gloucester, The Duke of Kent, Princess Alexandra of Kent, and Prince and Princess Michael of Kent all do. (Okay, those after Prince Edward don't do something every week - but they also only get a little money for the things they do.)
If we were to have a President, we would probaby have to pay nearly as much, and we wouldn't get all the glamour and family attached. (Just think who we might have had as president - usually former politicians - John Major, Margaret Thatcher, Paddy Ashdown, Michael Foot? Or possbily a trade unionist or two, such as Arthur Scargill? Or a film star - Peter Ustinov, Michael Caine, Albert Finney, Maggie Smith?) All countries have Heads of State and it seems to me that the Constitutional Monarchies of Europe provide an good solution, separating the politicians (Heads of Government) from the Heads of State.
On the younger members of the Royal Family, William and Harry are Army Officers, who have a few charities under their wings. Beatrice and Eugenie are still teenagers, and would not be expected to take on duties so young. Peter and Zara Phillips are not Royal - they have a lot of money (but no-one seems to complain about this) and lead their own lives.
Yes, some of them seem to be at a lot of parties - but their peer group go to a lot of parties. Come to think of it, when I was their age, I went to a lot of parties. My children went to a lot of parties. The difference is that we did not have a lot of paparazzi waiting outside to take photos of us. I think we all might have occasion to blush if we had. I think the ages of the Princes/Princesses are between about 25 and 17. If you can't spend some of your evenings and weekends socialising when in that age group, you would be a very sad person. I feel sure that Beatrice and Eugenie will take on royal duties, but probably not until they are some years older.
And, if Sarah, Duchess of York, wants to give money to her daughters, why shouldn't she? Since her divorce, she has received no public money. She has worked very hard and earns good money. Like most mothers, she probably doesn't keep it all to herself, but provides many things for her girls.