Is the "expansion of space" limited to intersteller space only? Does the expansion of space (that started from BigBang) limited to expansion of intersteller space only (as observed by Hubble) or does it happen even in the local observable universe? 
Specifically, does the space between two stars in our own galaxy is also expanding? Does the molecular space in an ordinary everyday object expanding as well (even if it is very very tiny)?
 A: This one is tricky because we have to admit here that the expansion of space is a model.  We don't actually know whether it is the true nature of the universe, or merely a good predictive model of every observation we have made over the course of human history.
That being said, the current preferred model is that the expansion of space occurs everywhere.  It is treated as an aspect of the topology of space itself, not just the matter in it.
Given the current estimate for Hubble's Constant of 71.9 km/s / Mpc, we can plug in numbers to determine how fast two points are moving apart.  Let's pick a point in Los Angeles and one point in New York.  They are 10083km apart (straight line distance through the Earth).  That's .00000000000000032676778 Mpc, if you convert km to Mpc.  Thus New York and Los Angeles are moving apart due to the expansion of space at a rate of 736 um/yr!
Of course, plate tectonics causes shifts 3 orders of magnitude greater than this, so we don't notice this expansion all that much.
A: According to modern understanding of physics, no.
In fact, if the accelerating expansion continues for trillions of years, space will be expanding so much (everywhere) that it will exceed the weak and strong force and pull everything apart.
EDIT
Alex Fillipenko (Seen a lot on "The Universe" is actually one of the leading experts in the area, so I wanted to include his lecture which he had given several dozen times all around the world from 2013-2016:
https://www.youtube.com/watch?v=CaCEEHtPcSg
