Do all lines have a y-intercept?

Answer 1

They do unless they're perfectly vertical.

If the equation of a line is [ x = a number ], then it has no y-intercept.

If 'y' appears in the equation at all, then the line has a y-intercept.

Answer 2

No. One easy example is a horizonal line like y=4. This line never crosses the x-axis, therefore it does not have an x-intercept.