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.
If two vectors are added, their arrows go in one direction, say clockwise around the triangle and the arrow on the resultant vector goes the other way (anti-clockwise).