No, the 7433 is a digital set, only compatible with the KX-TD series of systems or the more recent KX-TDA series. 

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.