I want to run a two-way ANOVA. I have unequal sample size and equal variance. Can I run the analysis?
If you are quite sure about the normal distribution of the data or have huge sample sizes, you can do.
However, if the data are rather not normally distributed, it is a good idea not to use the pooled variance estimators in the usual ANOVA. Instead, use variance estimators by cell and apply the Box Approximation (G.E.P. Box, 1954) for an approximative but more robust result.
If you choose your test procedure depending on a prior Goodness-of-fit-test (e.g. Shapiro-Wilk), you'll have a composite procedure that has different properties than the tests on the second step, so don't do this.
Yes, you can :) ANOVA doesn't assume equal sample sizes. It just assumes equal variances and normal distribiution in each group.
So you need to check normality (e.g. Shapiro-Wilk test) in each grup separately, or (easier) check normality of residuals.