Good evening Arnout and welcome to the Events Calendar Support forum!
Thank you for reaching out to us.
I would love to help you with this topic.
The first thing I would recommend (if you haven’t already) is to read our Themer’s guide.
If nothing else that will answer your second question (is it possible to separate date and time)
Once that’s done, I also recommend browsing through our functions Archive: https://theeventscalendar.com/functions/
To make a long story short, you can totally customize what is in each view. The extent of our support however will be limited to getting you started in the right direction as we unfortunately do not provide full support for customization.
So in other words, you are right, it is totally feasible to add filters to any page. In fact we have another add-on just for extra filters (although not the specific one you are looking for): https://theeventscalendar.com/product/wordpress-events-filterbar/
Let me know how that goes.
Best regards,
Geoff B.