Not at all. The significant interaction means that the effect of one variable is significantly different for each levels of the other variable. It has nothing to do with separate analyses. You should

**not** do any other tests, and all your two-way ANOVA and Bonferronis are completely correct.

The results won't differ in that case, and even if you give a significant result, it is likely a "false" positive error. But I've seen some studies in which thy have

*incorrectly* done two one-way ANOVAs.

No, the correct way is to do a '

**two**-way ANOVA' (regardless of existing an interaction or not), even when the results would be the same with running two One-way ANOVAs.

Even if one of the three P values is significant, you can run a Bonferroni.

You "can" also run a Bonferroni, when there is "no" significant ANOVA P value. It is recommended not to do so, because usually the Bonferroni will also give you nonsignificant results. But you "can" do it.