Are the limits due to the uncertainty principle and speed of light fundamentally the same phenomenon?

Nope h <> c.

h Planck's thing, is simply a constant of proportionality that came from E = hf; where a photon's energy is proportional to its frequency   This can actually be plotted on an E vs f where the slope is h.  It's that simple.  

As you can readily see h units are Joule sec/cycle.  Light speed c is in length/unit time.

Not clear what you mean by space and time being equivalent.  They aren't equivalent in any fashion I know of.  Space, the spatial dimensions, are separate and apart from the time dimension.  

We call our universe a 4 dimensional universe because it takes a minimum of four dimension (3 space, 1 time) to fix a position in our universe.  So space and time are related in that all four dimensions are necessary for that point.  But they are not equivalent in that we can't convert space into time, and vice versa.

There are other relationships, like how both space and time are affected by extreme gravity fields.  Space is bent and time is slowed; but, again, that does not make them equivalent...just related through gravity.