1. Do the scales of both of the axes on a graph have to start at 0?

Not necessarily, although the scales should be clearly marked.

In statistical graphs a zig-zag line is often used to indicate that a frequency scale does not begin at zero.

2. Should the dots on a coordinate graph be joined up?

This is optional.

If the number set being used is the whole numbers or integers (positive and negative) then dots are used. 
If the number set being used is the rational (fractions) numbers or real (all) numbers then a solid line should be used.