Engle-Granger approach to cointegration, and use of other integration tests The Engle and Granger define two series as cointegrated, if they are integrated, but a linear combination of the two series exist, which is stationary.
To estimate the cointegrating vector (i.e. the weights of the linear combination) they propose to estimate a regression with OLS and then test the residuals for stationarity.
In other words, if x and y are cointegrated, one estimates the equation
y = x + , thereby obtaining the integration vector , and tests  for stationarity. If there are more than two series, e.g. n series, there are n-1 cointegrating vectors.
To test integration of the residuals Engle and Granger use in the original paper the standard the Augmented Dickey-Fuller test, but Phillips and Ouliaris (1990) showed that these must be adjusted by the number of regressors. Later MacKinnon estimated more accurately those critical values, his last, most accurate estimation is in MacKinnon 2010 (whose critical values can be called for instance in the R package urca).
My question is: can one use other integration tests for the residuals (KPSS, Phillips-Perron, or any other), and if yes, why do everyone in the literature use use ADF (with MacKinnon critical values)?
If other tests may be used, should their critical values be adapted?
Which is the standard strategy for choosing the lag order of the ADF test? Which information criterion is the most used? Frequently this information is not mentioned at all.

Actually, you ask several questions. As to the first, yes. Phillips and Ouliaris (1990) is a cointegration version of Phillips-Perron, just as EG (who however also point out the need to take into account the number of regressors!) is a version of ADF. Shin (ET 1994), see https://econpapers.repec.org/article/cupetheor/v_3a10_3ay_3a1994_3ai_3a01_3ap_3a91-115_5f00.htm, proposed a KPSS type of test of cointegration.
