Will a TSO C129 GPS Navigator go into Approach Mode with an out of date datacard? Yes it will, it is a feature of the TSO based around a philosophy that it would be better in an emergency to be in the situation of having to fly an approach with an out of data card than to have an otherwise good navigation system that was unusable.
Does the Trimble 2101 go into DR mode when it loses signal? From memory yes, but it's been a while since I used one.
The T2101 was one of the first TSO C129 navigators available. It's pertinent to note there were a number of major software changes during the early deployment. Notable amongst several is that on some versions the RAIM annunciator would be illuminated when RAIM was available and on other versions the RAIM annunciator illuminated when RAIM was not available.