Duemilanove has one serial port - right?
Well it has one hardware supported serial port...
This port is shared with USB - right?
Well, it is CONNECTED to the FTDI chip - alas quite unbreakable - which has an USB connector. Note that this connection uses two serial 1k resistors. This is terribly important for all your experiments!!!
This port is the RX/TX pins - right?
Exactly, pins 0 and 1
RX/TX pins can not be connected to other serial device when sending program over the USB - right?
Of course not, the bootloader is trying to communicate with the IDE
RX/TX pins can be used when program is already stored and USB disconnected and board powered from alternative source -right?
Not really, the disconnection of the USB line and thus USB power does not stop the FTDI chip from working. So you can just leave it as is. The RCV line from the FTDI chip will be LOW without traffic, and - remember! - is connected through a 1k resistor. so you have a puul-down here.
But the terminal is always closed when program is sent. Probably because it would conflict with the transfer - right?
This has something to do with Windows... It is not possible to share a COM port, there is one owner only! Of course there could have been a more intelligent implementation, of re-establishing the monitors connection after the program has been updated automatically....
Is there way to keep arduino serial monitor open at all times?
It will stay open as long you do not reload a program.. So what exactly is the issue?