How could I perfom mail search (maybe using ZmMailApp.mailSearch) in zimlet. Searching would be subject and result - conversation I would like to open in a new tab. Any idea? Or any idea where I could find some useful documentation or tutorials?
Did you check the zimlet developers guide? That's a pretty good documentation source for developing zimlets. Also, there's a link to the Javascript API documentation, where you can find further information.
About your search question: I would suggest creation a SoapDocument for a "SearchRequest" and send that using the SendRequest method. Check the above documentation for further information.