I have a SharePoint 2010 Web Application in Windows Authentication mode. Can I directly migrate them to SharePoint 2013 Claims/Form based authentication Web Application? I am planning to use third party tool. If yes what tools are available?
If using the database attach method to migrate content from a 2010 farm to a 2013 farm, then you'll want to either convert the 2010 web application to claims mode before the move, then migrate the databases to a claims mode 2013 web application (this is Microsoft's recommendation), or create a classic mode web application using PowerShell in 2013, attach the databases from the 2010 farm (still in classic mode), then perform the upgrade to claims mode in 2013. You should not attach a classic mode 2010 database to a claims mode 2013 web application.
Alternatively, as you mention, you can migrate the content using third part tools, for example AvePoint (DocAve Migrators) and Metalogix (Content Matrix) both provide migration tools to allow content to be migrated between versions of SharePoint.