If firms are associated with one country, then if you have firm fixed effects you don't need country dummies as well. In fact, you can't estimate both, since the country effect (unless interacted with a time dummy) is time invariant, so it is collinear with the firm fixed effect. Thus, you can't estimate both. But that is not a problem, because the country effect is already captured by the firm fixed effect.
Should you include year fixed effects? Depends on your data and research question, but if you want to control for year effects that affect all firms in all countries, then you should include them. For example, if there were a global macroeconomic shock in a year, then year fixed effects would be one way to control for it.
When you implement fixed effects in nonlinear panel models like logit, you shouldn't do it by throwing in dummies for firms and years like you might with OLS. Those estimates are biased if you have insufficient observations per dummy (you would have 11 observations per firm dummy, not enough). Instead, use the conditional logit fixed effects estimator, which should be implemented in newer versions of statistics software. In Stata, you can do this via
xtset firmid year
xtlogit depvar x1 x2 x3, fe
In short, you should use firm fixed effects if you believe you have not included essential time invariant explanatory variables. Fixed effects will control for those time invariant factors. You should not use fixed effects if you want to estimate the effect of particular time invariant factors. You could not estimate those coefficients jointly with fixed effects. In most cases, fixed effects make your regression more robust, and that's why most economists use fixed effects.
Last point: are you estimating bankruptcy probabilities of firms? It seems to me that survival models, rather than binary choice models, would be more appropriate for this question.