Do two linear equalities always have a solution? If not, why not? If
yes, under what conditions?

For two linear equations of the form

y = (m1)*x + b1
y = (m2)*x + b2

there are three cases.

(1) The slopes are different m1 ? m2.
    The two lines intersect in one point.
    There is a unique solution.

(2) The slopes are the same m1 = m2 and the y intercepts are different b1 ? b2.
    The two lines are parallel.
    There is no solution.  

(3) The slopes are the same m1 = m2 and the y intercepts are the same b1 = b2.
    The two lines are coincident (i.e. the two lines lie on top of each other).
    There are an infinite number of solutions.

Also, if the linear equations are not limited to the 2 dimensional
field, there is another possibility:

The 2 lines may be skew lines (Or lines that are not parallel yet do not intersect)

To immagine this take 2 pencils and pretend that they are lines. Put
one pencil on a flat surface such as a desk and hold the second pencil
in such a way so as every point along the pencil is 2 inches above the
flat surface. You will notice that no matter what direction you turn
your pencil, it will never touch the second pencil, yet will only be
parallel in 2 situations.