Why is the voltage across the capacitor constant?
1) The simulator starts by calculating a DC solution of the circuit. In the DC solution capacitors are charged to the voltage that would be on a node after a very long time. The transient (time) simulation you performed starts calculating at time = 0 with this DC solution. So that is the solution where the capacitor is already charged.

If you want to see the charging behaviour of the capacitor, you must convince the simulator to calculate a DC solution where the voltage across the capacitor is zero. In some simulators this can be done with an "initial condition".

However it is much simpler to replace your DC voltage source with a pulse-source. Let it make a pulse that starts at 1 second and ends at 100 seconds. Then you will see the charging curve.

2) You're not charging the capacitor so that is why. If you were you would see that predicted current briefly when the voltage across the capacitor is zero.