1. Is the direction of the arrow on a vector important?

Yes, the direction of the arrow is vital.
If the arrow is pointing the wrong way this represents a completely different vector.

2. Does a vector have to have an arrow?

Yes. Without an arrow a vector becomes just another line.

3. In a vector triangle do all of the arrows point around in the same direction?

No, one arrow will always go in a different direction.
e.g. Two vectors will be clockwise and the resultant will be anti-clockwise.

4. For bearings, where does the angle measurement start?

A direction of due North is a bearing of 0°.