Let me add the comment that in the article of Sebastião  e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any  locally integrable function is a distribution--take the distributional derivative of its primitive.  Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$.  In a similar manner, any rational function can be regarded as a distribution.  The details of all this is in the article I cited.

b)  within this framework the concept of limits of distributions at finite of infinite points, values of distributions at a point and integrals of distributions are developed.  Although a general distribution doesn't have a value at a point, most ones which occur in practice DO at many points and this is significant.