When I connect phone to computer via USB cable, I can browse internal memory and SD card. Is there any way to refresh files/folders without replugging the device?
Unfortunately this doesn't seem possible. Even restarting Explorer doesn't work. I think the problem is that the phone sends the data to the PC when it's first connected, and after that just doesn't feel the need to update the PC. You can observe this by unplugging and replugging the phone really fast; it won't update because the phone didn't recognize that it was disconnected before you reconnected it again.
Seems like for now it's true what Shawn has said - the Phone send data to PC when it's first connected.
Apart from plugging in and out, you can obtain the same effect without touching the USB cable - open device manager, find your phone in Removable devices, then right click on it and choose Turn off, then Turn on. The effect will be the same as with replugging.
As it works with device manager, you should be also able to do it with PowerShell and/or Devcon (though this way is probably for developers rather than normal users).