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1. There may indeed be some transformation of your data that produces an acceptably normal distribution. Of course, now your inference is about about the transformed data, not the not-transformed data.

2. Assuming you are talking about a oneway ANOVA, the Kruskal-Wallis test is an appropriate nonparametric analog to the oneway ANOVA. Dunn's test (not the garden-variety rank sum test) is perhaps the most common nonparametric test appropriate for post hoc pair-wise multiple comparisons, although there are other tests such as the Conover-Iman test (strictly more powerful than Dunn's test after rejection of the kruskal-Wallis), and the Dwass-Steele-Crichtlow-Fligner test.

3. Multiple comparisons procedures (whether family-wise error rate variety or false discovery rate variety) don't really have anything directly to do with your specific test assumptions (e.g., normality of data), rather they have to do with the meaning of $\alpha$ (willingness to make a false rejection of a null hypothesis) given that you are performing multiple tests.

The ANOVA is based on a ratio of within group and between group variances. I am not entirely sure what you mean by heteroscedasticity in this context, but if you mean unequal variances between groups, that would seem to me to fundamentally break the logic of the test's null hypothesis.

A simple Google Scholar query for "Dunn's test" along with a general term from your discipline should return plenty of published examples.

References

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.

1. There may indeed be some transformation of your data that produces an acceptably normal distribution. Of course, now your inference is about about the transformed data, not the not-transformed data.

2. Assuming you are talking about a oneway ANOVA, the Kruskal-Wallis test is an appropriate nonparametric analog to the oneway ANOVA. Dunn's test (not the garden-variety rank sum test) is perhaps the most common nonparametric test appropriate for post hoc pair-wise multiple comparisons, although there are other tests such as the Conover-Iman test (strictly more powerful than Dunn's test after rejection of the kruskal-Wallis), and the Dwass-Steele-Crichtlow-Fligner test.

3. Multiple comparisons procedures (whether family-wise error rate variety or false discovery rate variety) don't really have anything directly to do with your specific test assumptions (e.g., normality of data), rather they have to do with the meaning of $\alpha$ (willingness to make a false rejection of a null hypothesis) given that you are performing multiple tests.

The ANOVA is based on a ratio of within group and between group variances. I am not entirely sure what you mean by heteroscedasticity in this context, but if you mean unequal variances between groups, that would seem to me to fundamentally break the logic of the test's null hypothesis.

A simple Google Scholar query for "Dunn's test" along with a general term from your discipline should return plenty of published examples.

References

Conover, W. J. and Iman, R. L. (1979). On multiple-comparisons procedures. Technical Report LA-7677-MS, Los Alamos Scientific Laboratory.

Crichtlow, D. E. and Fligner, M. A. (1991). On distribution-free multiple comparisons in the one-way analysis of variance. Communications in Statistics—Theory and Methods, 20(1):127.

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.

1. There may indeed be some transformation of your data that produces an acceptably normal distribution. Of course, now your inference is about about the transformed data, not the not-transformed data.

2. Assuming you are talking about a oneway ANOVA, the Kruskal-Wallis test is an appropriate nonparametric analog to the oneway ANOVA. Dunn's test (not the garden-variety rank sum test) is perhaps the most common nonparametric test appropriate for post hoc pair-wise multiple comparisons, although there are other tests such as the Conover-Iman test (strictly more powerful than Dunn's test), and the Dwass-Steele-Crichtlow-Fligner test.

3. Multiple comparisons procedures (whether family-wise error rate variety or false discovery rate variety) don't really have anything directly to do with your specific test assumptions (e.g. normality of data), rather they have to do with the meaning of $\alpha$ (willingness to make a false rejection of a null hypothesis) given that you are performing multiple tests.

The ANOVA is based on a ratio of within group and between group variances. I am not entirely sure what you mean by heteroscedasticity in this context, but if you mean unequal variances between groups, that would seem to me to fundamentally break the logic of the test's null hypothesis.

A simple Google Scholar query for "Dunn's test" along with a general term from your discipline should return plenty of published examples.

References

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.

1. There may indeed be some transformation of your data that produces an acceptably normal distribution. Of course, now your inference is about about the transformed data, not the not-transformed data.

2. Assuming you are talking about a oneway ANOVA, the Kruskal-Wallis test is an appropriate nonparametric analog to the oneway ANOVA. Dunn's test (not the garden-variety rank sum test) is perhaps the most common nonparametric test appropriate for post hoc pair-wise multiple comparisons, although there are other tests such as the Conover-Iman test (strictly more powerful than Dunn's test after rejection of the kruskal-Wallis), and the Dwass-Steele-Crichtlow-Fligner test.

3. Multiple comparisons procedures (whether family-wise error rate variety or false discovery rate variety) don't really have anything directly to do with your specific test assumptions (e.g., normality of data), rather they have to do with the meaning of $\alpha$ (willingness to make a false rejection of a null hypothesis) given that you are performing multiple tests.

The ANOVA is based on a ratio of within group and between group variances. I am not entirely sure what you mean by heteroscedasticity in this context, but if you mean unequal variances between groups, that would seem to me to fundamentally break the logic of the test's null hypothesis.

A simple Google Scholar query for "Dunn's test" along with a general term from your discipline should return plenty of published examples.

References

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.

1. There may indeed be some transformation of your data that produces an acceptably normal distribution. Of course, now your inference about about the transformed data, not the not-transformed data.

2. Assuming you are talking about a oneway ANOVA, the Kruskal-Wallis test is an appropriate nonparametric analog to the oneway ANOVA. Dunn's test (not the garden-variety rank sum test) is perhaps the most common nonparametric test appropriate for post hoc pair-wise multiple comparisons.

3. Multiple comparisons procedures (whether family-wise error rate variety or false discovery rate variety) don't really have anything directly to do with your specific test assumptions (e.g. normality of data), rather they have to do with the meaning of $\alpha$ (willingness to make a false rejection of a null hypothesis) given that you are performing multiple tests.

The ANOVA is based on a ratio of within group and between group variances. I am not entirely sure what you mean by heteroscedasticity in this context, but if you mean unequal variances between groups, that would seem to me to fundamentally break the logic of the test's null hypothesis.

A simple Google Scholar query for "Dunn's test" along with a general term from your discipline should return plenty of published examples.

References

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.

1. There may indeed be some transformation of your data that produces an acceptably normal distribution. Of course, now your inference is about about the transformed data, not the not-transformed data.

2. Assuming you are talking about a oneway ANOVA, the Kruskal-Wallis test is an appropriate nonparametric analog to the oneway ANOVA. Dunn's test (not the garden-variety rank sum test) is perhaps the most common nonparametric test appropriate for post hoc pair-wise multiple comparisons, although there are other tests such as the Conover-Iman test (strictly more powerful than Dunn's test), and the Dwass-Steele-Crichtlow-Fligner test.

3. Multiple comparisons procedures (whether family-wise error rate variety or false discovery rate variety) don't really have anything directly to do with your specific test assumptions (e.g. normality of data), rather they have to do with the meaning of $\alpha$ (willingness to make a false rejection of a null hypothesis) given that you are performing multiple tests.

The ANOVA is based on a ratio of within group and between group variances. I am not entirely sure what you mean by heteroscedasticity in this context, but if you mean unequal variances between groups, that would seem to me to fundamentally break the logic of the test's null hypothesis.

A simple Google Scholar query for "Dunn's test" along with a general term from your discipline should return plenty of published examples.

References

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.