Is it legal to publicly disclose the information of received DMCA takedown request including user information of requester?
Yes.
which included untrue information.
And the requestee decided to publicly disclose information of the requester including his username on the platform, with extra insulting remarks, and additional information that he had filed a counter-notice against DMCA takedown request. Is there any legal action the requester can take against that?
False statements of fact that damage someone's reputation are sometimes actionable in a defamation action. There are a great many factors that go into the full analysis of when defamation lawsuits can prevail. Usually, one can only prevail if the false statement of fact was knowingly fraudulent and if one also suffers provable economic harm as a result of that statement. Statements made in connection with a formal litigation process are often not actionable at all outside a request for sanctions in the context of the litigation.
Mere insults are not actionable in the U.S. unless they are part of a larger course of conduct that amounts to harassment, and not alway even then.
True information and statements of opinion, even if insulting, are not actionable.