Good evening Fabio and welcome to the Events Calendar forum!
Thank you for your interest in our products.
We’ll be glad to answer your questions.
– if we do “do_shortcode()” does it work in a page template?
Yes, absolutely.
– if we navigate througout the months, will the menu remain opened as in the image attached?
It looks like you will probably want to use our Pro widgets shortcodes: https://theeventscalendar.com/knowledgebase/pro-widget-shortcodes/
The mini-calendar widget seems perfect for that purpose.
As for how it looks, you can tweak it as much as you want.
To do that you simply need to use a template customization of the view you want to modify.
You might want to read our Themer’s guide to get a sense of how that works.
– can we call the shortcode via ajax?
If you have a way to execute normal WordPress shortcodes via Ajax, then these shortcodes should work as well.
Let me know if that helps.
Have a great day!
Geoff B.