Bicarbonate

You are talking about different situations:

1) You dissolve a carbonate salt in water to form an aqueous solution. The solution of carbonate (CO3^2-) or bicarbonate (HCO3^1-) salts is slightly alkaline due to the presence of the hydroxide ion.

2) strictly speaking, H2CO3 does not exist. This equation actually means dissolving carbon dioxide in water. Carbon dioxide is an acidic gas and it will give H+ in the resultant solution.

**You asked about the nitration using con sulphuric and nitric acid, supphuric acid does not act as a catalyst at all, it is an oxidising agent in this reaction and give out nitronium ion (+NO2) which is added to an electron rich area like benzene ring

Is there two question?

In eqn 1,2, they are just equilibria.
Surely, acid can react with alkali. But they can both exist.
Just like in water, there are H+ and OH-. It is the dissociation of water.

In the case of carbonate, carbonate react with water and form those products.
They all exist in the solution.

What make you confused is the neutralization bteween them. In fact, equilibrium is a dynamic process, forward and backward reaction are occuring at the same rate.