How to detect if Ergodicity, Stationarity and Martingale. dif. sequence are satisfied?
Ergodic and strict stationarity are the essentially the weakest assumptions for which you have a LLN, i.e. can do large sample estimation. Given the very liberal way applied econometricians use laws of large numbers, ergodicity and strict stationarity is almost always assumed. If that is in question, you're simply not in business for consistent estimation.

Similarly, if one would like to have a CLT, i.e. do large sample inference, MDS is a general weak assumption. (There are other CLT's around but more technical. The MDS CLT and Lindenberg CLT are the two immediate generalizations of the classical CLT.)

The MDS assumption can be tested. An MDS is just a discrete-time martingale with mean zero. In particular, its increments are uncorrelated. (The martingale assumption is an intermediate condition between serially uncorrelated and independence.) So a rejection by a serial correlation test would also reject the MDS assumption.