Understanding capacitive current
(1) I'd say the notation \$I={Q \over t}\$ is confusing. As the Wikipedia article states it:


  An electric current is the rate of flow of electric charge past a point or region.


Right there we should key on the phrase rate of flow. And it's pretty intuitive too -- when we think about current, it is simply moving charge from one place to another, so all that matters is the change in the amount of charge, not the absolute amount. So I'd say it would be more accurate to write \$I={\Delta Q \over \Delta  t}\$, but in practice people don't do this. At any rate, in this context, \$I={dQ \over dt}\$ should hopefully make perfect sense.

(2) Simply because the capacitor value is constant. If you replaced the constant \$C\$ with a number (say, 2.5), then seeing \$2.5 * V = Q\$ differentiated into \$2.5 {dV \over dt} = {dQ \over dt}\$ would make perfect sense. So just remember that \$C\$ is a constant, not a time-varying quantity.

(3) It is the current flowing through the capacitor, in one terminal and out the other. (Sorry for the... unreasonable... size of the schematic image...)







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