And if the unit is set to TRANSMIT, will it send the PC as soons as it receives a PC? Or do I need to transmit the PC manually after the H9 has received the PC?

Various answers here. If you have recent software which has the "XMT.PC" command on the System MIDI menu, it will send a PC (via the TX map) on receiving one if enabled.

Otherwise you could set OUTPUT to MERGE, which combine THRU and XMIT. Note that this is unsuitable for heavy MIDI traffic.