It's packaged in Debian and Ubuntu as "doublecmd-qt" and "doublecmd-gtk", so Linux Mint ought to have it as well.
> Can it filter-as-you-type the file list?
Yes.
> Does it have chainable script button bars?
I don't know what that is.
> Is there a poweful rename tool?
There's a Multi-Rename Tool, yes.
> Can it look pretty
You'll have to be the judge of that. It looks good to me. It uses the Lazarus component library, so it can be built with either Qt or GTK (or native win32 on Windows) and uses whatever themes and configuration applies to those.