Could TCAS be used as a backup source for position, attitude or air data?
Can TCAS assist in dead reckoning?

First, understand that TCAS runs off secondary (transponder) radar data. Essentially your aircraft with TCAS is like ATC radar, interrogating other transponders, getting their pressure altitude and getting their bearing from your aircraft.  That data combined is processed to create threat advisories.

If displayed on a moving map, the moving map is typically fed positional data for your aircraft from GPS, and the TCAS alert data is overlayed onto the map.

So to address your question, TCAS only provides data about transponders relative to your aircraft.  TCAS does not determine where you are, or where you are going. (Depending upon your display it might appear that way, but that is not part of the TCAS system.)

So the answer to your question is no, TCAS does not effectively create a secondary navigation capability.

Addendum #1
TCAS can only sense altitude through the host aircraft's static port, so if it is iced over, TCAS has invalid altitude information.

TCAS does not sense other air data, such as pitot pressure information (airspeed).

TCAS does not know the host aircraft's position.  Any positional information in TCAS is derived from pressure altitude information the host and all interrogated transponders.  If the host static port is obscured, the information on threats will be invalid, because TCAS displays relative to the host pressure altitude.

In other words, there is no absolute positional information, and all TCAS information is relative to the current host aircraft, and derived from the secondary beacons of interrogated aircraft transponders.