Why do both plates of a capacitor have the same charge? How do we know that both plates of a capacitor have the same charge?
You could argue conservation of charge, but I don't see how conservation of charge implies the charge on both plates is the same.
Say you have a charge of +q on one plate and -m on the other. +q doesn't equal -m.  How do we know that the difference of charge doesn't end up on the terminals? Isn't that possible? Wouldn't conservation of charge not be violated if the charges on both plates are not equal, but the difference in charges ends up as surface charge in the wire or some other place in the circuit?
 A: There's no reason the sides have to be equal, but if they aren't, the capacitor obviously has a net electric charge. Moreover, the electric field lines emanating from the capacitor have to go somewhere, such that the whole capacitor is also one half of a larger capacitor. In a circuit model, you would simply represent this as two or more separate capacitors, each individually balanced with zero net charge.
If the net charge of the entire circuit is nonzero, then you have to add a capacitor with a terminal going "nowhere," to a node representing the outside world. The capacitance of this depends on environmental conditions, i.e. the dielectric constant of air, and the shape of the electric field (particularly the surface area of the exposed metal).
A: 
How do we know that both plates of a capacitor have the same charge?

In the context of ideal circuit theory, KCL (based on conservation of electric charge) holds.
For a capacitor connected to an external circuit, KCL demands that the current into one terminal equals the current out of the other terminal.  This implies that the charge on each plate is equal and opposite.
Now, it is certainly possible to place unequal charge on the plates of a capacitor and I've seen this done in an undergrad physics lab.  But it wasn't in a circuit context.
Context is crucial.  Attempting to apply results outside of the context (assumptions) upon which they're based is an elementary (though, unfortunately, common) error.

Addendum to address a comment by @Physiks lover:

user41086's point is a good one. KCL is concerned about currents at a
  single node, not at multiple nodes.

It isn't a good point because the statement that KCL isn't concerned about current at multiple nodes isn't true.  One can draw a surface enclosing two or more nodes and KCL holds for the supernode.
Supernodes are most commonly used to enclose a floating voltage source in order to apply node voltage analysis.

But supernodes are more general than that.  For example, see these MIT EE course notes on nodal analysis.

"The part of the circuit enclosed by the dotted ellipse is called a
  supernode. Kirchhoff’s current law may be applied to a supernode in
  the same way that it is applied to any other regular node. This is
  not surprising considering that KCL describes charge conservation
  which holds in the case of the supernode as it does in the case of a
  regular node."

Thus we can enclose the entire capacitor with a supernode and apply KCL.  In this ideal circuit context (which includes a number of assumptions that only approximate reality), the stipulation that the charge on each plate is equal and opposite is valid.
A: Capacitor needs to be connected to some dc source for completely charging it.When you connect it to the DC SOURCE, the current will start to flow such that charges on both surface is increasing with opposite polarities. To obey KCL, charges on both plates should must be same. WHICH IS(That is KCL) TRUE HERE DUE TO AMPERE'S Modified Circuital law which includes displacement current.
If Suppose you just bring charged plate near another neutral plate then charge of opposite polarity is gathering on near side of neutral plate(Still neutral).Then you just connect other side of neutral plate to earth to get overall negative charge (which may not be equal of another positive charged plate) and earth accepts the positive charge from plate which is now negatively charged and doesn't show much change in potential due to high capacitance of 711 microfarad.So overall charge is still conserved.
