It can't work on your current system at all. The KX-T123211D is not a digital system, it is analog.
The only phones that can be used with your system are the KX-T70XX and 77XX series. There are some much older phone models that are also supported, but I doubt you will be interested in their appearance.
To answer your second question, yes, you can connect a modem directly to any extension jack and dial 9 to get an outside line.
Depending upon how your system is set up, you may be able to share an existing jack with a multi-line phone and the modem using a basic splitter, but only one can be used at a time.