What percentage of publicly traded companies that undergo lawsuits make it out without going bankrupt?

Question

I am referring to lawsuits that involve the Board of Directors, higher-ups, etc.

Answer 1

Nearly all. Most lawsuits involving Board of Directors and corporate officers are usually settled out of court. Most, if not all, public companies buy a special kind of insurance called Directors and Officers Liability Insurance (or simply D&O), which pays for those settlements.