The Tukey-Kramer Test is a Tukey Test for unequal sample sizes. Most software packages use it when you ask for "Tukey's Test". Using multiple t tests results in an inflated Type I Error rate.
Thank you for your answer ! I'm working with GraphPad Prism, I guess it computes the Tukey-Kramer automatically when needed (no notion of the Kramer correction, but no objection to calculation from the logiciel neither when a subject is missing, so...).
If you use multiple post-hoc tests without a built-in experimentwise error rate, you should use a Bonferroni adjustment or the one by Sidak so that the experimentwise error rate stays controlled. Using the K-W Test versus the F Test depends on the data and the assumptions regarding normality and equal population variances. Also, the K-W Test looks at equality of distributions, which is "deeper" than testing for the equality of population means.
I agree with the explanation by Raid Amin. ANOVA with multiple comparisons are designed to compare more than two groups and to avoid the type I error problem. I believe there are many options for the multiple comparisons (posthoc test) and one of them should fit your situation of non equal sample sizes across groups. If you really have to do multiple t-tests, at least you need to adjust the p-values based on Bonferroni.