dimensional analysis - Deriving or justifying fundamental constants

Is there a fundamental way to look at the universal constants ? can their orders of magnitude be explained from a general points of view like stability, causality, information theory, uncertainty?

for example, what sets the relative magnitudes of Planck's constant compared to say charge of electron or is it just a matter of choice of units.

Does the physics become more or less cumbersome, insightful if we set all fundamental constants to 1 in appropriate units.

Answer

Your last couple of comments, about units, are incredibly important. It only makes sense to compare two things if they have compatible units. So, to use your example, it doesn't really make sense to talk about the size of Planck's constant relative to the electron charge, but it does make sense to talk about the mass of the muon relative to the mass of the electron.

There are a number of fundamental dimensionless constants in physics. Some are ratios of similar things, such as $m_\mu/m_e$. Others are not so obviously expressed in that form, such as the "fine-structure constant" $\alpha=e^2/(4\pi\epsilon_0\hbar c)$. It makes sense to wonder why the dimensionless constants have the values they do -- one could imagine that a future theory would explain these values. (As Georg says, there is no such theory right now -- by definition the "fundamental" constants are those that don't have a deeper explanation.) On the other hand, one would not expect any fundamental theory to explain why a dimensionful constant, such as the speed of light, had the value that it did: the fact that $c=2.9979\times 10^8$ m/s is an artifact of our definitions of the meter and second. (Actually just the meter in that particular case.)

To answer your last question, most people who work in fundamental physics do choose their units so as to set as many fundamental constants equal to 1 as possible. This makes the equations look cleaner, and also makes it easier to see what the interesting dimensionless constants are.

Sometimes people try to "explain" the values of various physical constants via anthropic reasoning, pointing out that if the values were quite different from what they are the Universe would be inhospitable to life. Other people find those arguments to be logically suspect or even morally reprehensible.