Can I charge it when I shut down?

For charging, it doesn’t matter how you set it up, after you shut it down.
Setting it as server, plugging it in, and not intervening will actually make it charge slower.
Server mode prioritises wifi, to prevent dropouts, BUT also keeps the clockwork pi on. The screen will turn off, but the machine is still active.

Also, @apple - check out the search function in the forums! You can find a lot of the answers to your problems, usually asked by other members. I found this one by typing “Power” in the search field.