Did you do an outlier review before the analysis started? On what criteria did you evaluate the normal distribution of the given data? Which technique did you use for nonparametric ANOVA? What is the method you use for binary comparisons?

Yes, it is possible. Whether or not it means anything depends very much on the experimental design and whether the ANOVA was the correct statistical test to use. In general, if your ANOVA results were non-significant, you should not be doing a post-hoc analysis. Significant results will appear by chance.

It depends on the post hoc procedure you use. A few require overall significance of the omnibus F test for the effect to proceed. Examples include Fisher's LSD. However, these procedures may not be very appropriate because they typically do not protect against partial null hypotheses (only the complete null hypothesis). Thus they don't fully protect against Type I error inflation. In practice if you have only three means Fisher's LSD is a reasonable choice (as there can only be on fall partial null hypothesis). With 4 or more means Type I error rates can rise  (in some cases sharply) above nominal alpha.

Many post hoc tests (probably the vast majority) don't require overall significance of the omnibus F test for the effect to proceed. If you only use these corrections when the omnibus F test is statistically significant then this can dramatically decrease statistical power as you over-correct alpha per test. A good example is Tukey's HSD which protects against the complete null hypothesis and all partial null hypotheses.

I just noticed that you are using non-parametric ANOVA of some kind. Hard to say but likely there is no correction for Type I error. I suggest using a general correction such as the Holm or (better still) Hochberg that just uses the raw p values. This is trivial to do in R using the p.adjust function. It can also be done fairly simply by hand with a little work.

I think you should be more clear on which test and posthoc test you applied and if you applied any fwer correction when you make posthoc comparisons.

Excellent answer from Thom. If you need any help on the topics he mentioned you should have a look at the textbooks by Rand Wilcox "Introduction to robust estimation and hypotheses testing" or "Modern statistics for the social and behavioral sciences". Both give a good overview of the above mentioned and other methods and furthermore R codes for all analyses.

If we are not getting the significant results by using ANOVA, we can't get significant results in post hoc tests.

@Subhash: of course we can, but we should not interpret them (under normal circumstances). And as Thom already mentioned, not all tests even require a significant overall F-Test.

I would add that Thom's answer is valid when you have "planned comparisons" prior to conducting the statistical test. It is not clear from the wording of your question whether this was the intent. With planned comparisons you will be looking at the results only for those comparisons, and ignoring all others (significant or not). If you did not have planned comparisons, then a great deal more caution is needed.