If all aspects of a test are specified in advance and its assumptions are met, you can safely conclude that the null hypothesis will be rejected erroneously at the frequency defined by the error level. If you conduct several tests (a “family” of tests), each of these tests is an additional occasion to commit this error.
Each individual test might still have its nominal error level but the probability that you reject at least one null hypothesis erroneously in the family will be higher. To the extent that you had reasons to set an error level in the first place, this is a problem as the probability to commit at least one error is higher than said error level. This is the core of the concern about multiple testing and it seems to apply to all four situations you describe.
Now, if the tests are independent and all null hypotheses are true, you know what the probability of committing at least one error over the whole family is (incidentally, you also know that any rejection must be erroneous). If they are not independent or some of the null hypotheses are in fact not true, not only is the actual family-wise error level higher than the nominal level but it's difficult to know exactly how high (you can however put bounds on it; that's the reasoning behind the Bonferroni adjustment). If the various hypotheses are related in some way, specific solutions might apply (for example classical “multiple comparison” techniques, multivariate tests, sequential procedures in clinical trials) but even if they do not, the problem is still there.
Repeatedly testing as you collect data (also known as optional stopping or “sampling to a foregone conclusion”), trying various techniques, analyzing various subsamples or dependent variables also expose you to multiple testing issues. These situations are not always discussed together but there is no reason why they should not. Different techniques testing the same hypothesis or related ones (your point 4) might be closely related and possibly not entail as much of an increase in the family-wise error level as multiple tests on completely unrelated samples but you are still conducting several tests.
Possibly the most delicate issue is point 3. In such a setting, you could very well run a single statistical test. How could that lead to a multiple testing problem? One argument in favor of this view is that the p-value depends on the distribution of a test statistic over hypothetical replications. If you would replicate this experiment, you would carry out a different test each time depending on how the data “look”. The distribution of this test statistic would not be the same as if you would blindly test the same comparison every time as it is also influenced by this prior informal visual inspection of the data. In fact, you are implicitly considering many possible comparisons in your study, a multiple testing situation.
A similar reasoning also applies to the situation described in point 4. It might or might not correspond to what is usually called “multiple testing issues” (the perennial problem of is-this-really-called-X questions) but the consequence is the same: The tests are uninterpretable as they could be far from their nominal error level. The situation is further muddled by the fact that you propose conducting further tests based on the results of the earlier ones but you are in any case willing to run multiple tests. (Note that this is based on the fact that you claimed to make decisions based on significance alone. Picking a model based on the residuals or some other diagnostics and only conduct one significance test seems a much better approach.)
My reasoning on the last two points is inspired in particular by Wagenmakers, E.-J. (2007). A practical solution to the pervasive problems of p values. Psychonomic Bulletin and Review, 14 (5), 779-804.