Is Princess Madeleine related to Queen Elizabeth II?

Yes, through two sons of Queen Victoria: Prince Arthur and Prince Leopold.

The Swedish Royal family are descended from the house Saxe Coburg - Gotha as are the British Royal family

They are all inter-related. It's why they have an unusual number of toes.

Yes.

Yes, Apparently she is the Queen's great, great, great, great, great, great, great, greZZZZZzzzzzzzzzzzz GREAT, GREAT Grandfather, three and a half times remove on her father's mother's great uncle's side. Just a minute, I'll draw the family tree for you, would you believe it, I have no lead in my pencil...what? WHAT?

Yes, in at least two ways actually. Princess Madeleine's paternal grandparents were both descendants of Queen Victoria, Queen Elizabeth's gg-grandmother:

Queen Victoria

Edward VII

George V

George VI

Elizabeth II

Queen Victoria

Prince Arthur

Princess Margaret

Prince Gustav Adolf

Carl XVI Gustaf

Princess Madeleine

Queen Victoria

Prince Leopold

Prince Charles Edward

Princess Sibylla

Carl XVI Gustaf

Princess Madeleine

This makes Elizabeth and Madeleine third cousins once removed twice over.

Yes, distantly.

Which Princess Madeline?

Yes. She's a third cousin once removed of Queen Elizabeth II.

Queen Victoria was her great-great-great-great grandmother.

Well, all european royal famillys probably are related in one way or another going back enough in time.

Queen Elizabeth II is the great-great-great-great granddaughter of King George II, and Madeleines father King Carl XVI Gustaf of Sweden is his great-great-great-grandson.