What happens when a conducting plate is grounded? (1) When I ground a conducting metal plate, is all the charge on the plate transferred to earth? By this I mean charge on surface of the plate and charge inside the plate.
Inside metal plate there are free electrons. So all the free electrons are transferred to earth? If yes, then metal plate should become positively charged.
On its surface, then it should have some constant non-zero potential. But when we ground conducting plate, potential on its surface should be zero.
(2) When we put positive charge on a metal plate, what do we put actually?
In nature protons are positive charged particles and electrons are negatively charged particles (apart from other elementary particles).
 A: Firstly, only excess electrons will move to ground. For example, if plate has n protons and n+x electrons then x number of electrons will move to ground so that number of protons and electrons become equal and the plate becomes neutral. Charge does not becomes zero, but "Net Charge" becomes zero. You may think what if it has n-x electrons. Then x electrons will move from ground to plate. 
Coming to your second question, if we charge the plate positively, then we don't put excess protons. We just remove electrons so that number of protons becomes more than number of electrons and the plate has 'Net' positive charge.
A: In electrostatics, when two conductors are connected to each other, their charges redistribute such that their potential difference is minimized. (Ideally the difference should be zero.) This is the most stable state for a system of conductors. 
Theoretically, since the Earth is usually taken to be at zero potential, a grounded conducting plate would be at zero potential too. (This is a general convention.)
For question (1), no, not all the charges from a conductor would transfer into the Earth.
Also, I think the answers here would help for (2) How to make something charged using electricity?
