Does a DC Current charge a DISCHARGED capacitor and does the current pass throug the capacitor?
The answer is kinda. The current will "flow" through the capacitor and go through the resistor. What is happening is that as the capacitor is getting charged, the charge in the capacitor is displaced by the incoming electrons so it comes out the capacitor and flows through.
For better detail, you should read this answer I found on Quora.