In the openning sequence, Why was Audrey Hepburn wearing an ice bucket

It may have looked like an ice bucket, but it was a (mercifully brief) 60s revival of a hat originally popular in the "Roaring 20s". The hat style is called a "cloche", French for "bell" because of the bell shape.

I'm old enough to remember when Cleopatra went to the Temple of Ra to lead a few cheers.

I'm sorry but the hat is not a cloche. Yes they were popular in the twenties and thirties but they fit snug on the crown and usually gently flair out around the face framing it. When viewed from the side it does look like a bell, hence the name. The definitive feature is the tight fit on the head. This one looks Egyptian inspired or British Queen's Guard. But not a cloche.

Read up on "Pop Art", a highly influential style in the mid-60's . . . . so influential that it bled over from painting and into fashion (Audrey's hat) and movies (Modesty Blaise).

Her clothes in this comedy were apparently intended as a spoof of 1960s high style. There wasn't much she wore that didn't make me laugh, and I remember the '60s.

It looks like a helmet or something. Even though she's driving a car.

Definitely not a cloche like one commenter said.