Is it possible to use that plugin and export custom posts crated via Types plugin to specific XML format?
Yes.
I see that plugin saves all presets as a template. Is it possible to run that template via WP Cron and store export files on a server instead export as a file to local computer?
Yes, cron exports work the same as they do in WP All Import. You can read more here: https://www.wpallimport.com/documentation/recurring/cron/
By default the export files are stored on your server in wp-content/uploads/wpallexport/exports. You can automatically send your data to Zapier to send it wherever you like. You can read more here: https://www.wpallimport.com/2015/11/wp-all-export-zapier-send-your-data-to-over-500-apps/
Is there any way to filter export output? I need to replace some XML elements like <data> and <post> with different element names as well wrap some particular sections with additional XML elements.
Kind of. You can change the element names and pass your data through a custom PHP function, but right now you can’t alter the structure of the XML file (wrap particular section in additional elements). We have plans to implement features like that, but they aren’t ready yet.