[Sebastian Kripfganz]

1. If your core predictor is endogenous, it is hard to justify that the squared term is exogenous.
2. If you choose the second lag of an endogenous variable as an instrument for the first-differenced model, then any serial correlation of the error term will invalidate that instrument. This is irrespective of whether there is a lagged dependent variable or not. A lagged dependent variable in the model can help to remove the serial correlation from the error term.
3. Similar to point 1, if you have an interaction term between an endogenous variable and an exogenous variable (e.g. a dummy variable), then as a default I would typically still assume that the interaction term is endogenous unless you can come up with a convincing argument why it is not. I would not put too much trust in the overidentification test results. In the first place, you need to have a good theoretical argument for the classification of your variables.
4. I am sorry that the estimation with xtdpdgmm takes such a long time. Eventually, it should still work with such large data sets. Admittedly, it is much slower than xtabond2. The reason is that there is a trade-off between flexibility of the command and its computational efficiency. xtdpdgmm is intended to provide quite a good bit of additional flexibility over xtabond2. This comes at the cost of a few inefficient parts in the code. If you do not need the extra flexibility, you might be better off with xtabond2 when using such large data sets.

[Huaxin Wanglu]

It's sad no one can help...?