Bicarbonate decomposition and pKa
Carbonic acid is not necessarily more stable than the carbonate ion. 

Why carbonic acid ( p$\ce{Ka = 6.3}$ )  is not stable?

The reason for this is same as the one given for the instability of aluminium carbonate, iron (III) carbonate etc. That is, the carbonate ion has a highly polarisable electron cloud, which can be distorted by densely positively-charged species, such as the hydrogen ion, in this case. Thus, carbonic acid is not a very stable species and compared to the bicarbonate ion, which only has one densely-charged hydrogen ion distorting its electron cloud, it is definitely more unstable. Thus, the position of the 2nd equilibrium mentioned in the question should be to the left, favouring the more stable species.

Why is the bicarbonate ion ( p$\ce{Ka = 10.3}$ ) more stable than the carbonate ion  ?

This is due to there being a smaller negative charge on the bicarbonate ion and there being still 2 equivalent resonance structures to spread out the negative charge. Compared to the carbonate ion, which has twice the charge density but yet only 1 more resonance structure, the bicarbonate ion is more stable. Therefore, the position of equilibrium in the first reaction mentioned in the question should be to the left, not to the right. 

Therefore, in solution, the dominant species among the three, should be the bicarbonate ion. This should be the correct line of thought.