CableCARD

A cable card is inserted into a compatible device (TV, DVR, etc.) and allows reception/recording of cable tv signals without a Cox provided set-top box.  You can rent one from Cox but your TV must be compatible, not all are.

The cable modem has nothing to do with a cable card ... the modem provides high speed internet service.  You only need one modem and a router or a gateway which is modem/router in one device.  The "smart" services of your smart TV will connect to the router, normally using WiFi.  A router is capable of handling multiple devices, TVs, PCs, tablets, etc.

I little more info on what you are trying to do will help in answering your questions.

AllenP said:

A cable card is inserted into a compatible device (TV, DVR, etc.) and allows reception/recording of cable tv signals without a Cox provided set-top box.  You can rent one from Cox but your TV must be compatible, not all are.

No currently manufactured TVs have a CableCARD slot.  There were a few around 2007-2008, but they did not support a Tuning Adapter or Switched Digital Video.

If you want to use a CableCARD on multiple TVs the best solution is to get a TiVo Roamio and a TiVo Mini for each additional TV.

Or wait until this Fall when SiliconDust releases the HDHomeRun DVR