If I plug in the Pentax K-1 to a wall socket using the adapter, will it charge the battery if I leave the "on/off" switch "off"? And will it still charge the battery if I leave it "on", just at a slower rate??
If, by "using the adapter", you mean plugging in a USB cable to the camera's USB port, no. The USB cable is for file transfer only, and does not charge anything.
If, by "using the adapter", you mean plugging in an AC power adapter into the DC input port, then again, no. The adapter itself takes the place of the battery and bypasses it; it does not charge the battery.
No. Most DSLRs - actually, I know none that does - do not have an integrated charging circuit.