Yes they do.
Yes, but the numbers 1A, 5V to 4V and 1 second in your example just don't live in the same space-time continuum together at the same time.
The fundamental equation relating the voltage across and current through a capacitor is
I = C dV/dt.
This equation is the equivalent to Ohm's law for resistors. There is a similar relation for inductors:
V = L dI/dt.
If you let dV/dt = 1 V/s and C = 1 F then I = 1 A.
The current in my example is constant as the voltage on the cap changes. This make it valid to apply the above eqn. directly.
If a resistive load is used to discharge the cap, then the current will change as the voltage falls.
If you have a series RC circuit, the T = RC and 63% stuff can be derived from this equation combined with V= IR.
For a resistor in a series RC circuit you end up with
V(t) = Vo * (1 - e ^ (-t/RC)).
V(t) = Voltage on the capacitor as a fn. of time.
Vo = voltage source driving the R and C in series.
R,C = resistance and capacitance of the resistor and capacitor in series.
t = time (t = 0 is when the voltage source is applied to the circuit)
If you let t = RC you get
V(RC) = Vo * (1 - 0.367) = 0.632 Vo.