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What are sample size and number of events low limits for a proper application of McNemar's test with continuity correction? When must I use McNemar's exact test?

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McNemar's test is applicable with any sample size but if you have a low sample size then it is not appropriate to approximate the null distribution of the test statistic by a chi-squared distribution. In this case you would need to apply the formula for an exact test, which is given in the linked page. This exact test can be applied with low sample sizes, and the p-value will be accurate, but the test will not have much power (unsurprisingly).

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