Are there possible ways to customize calendar in Saleforce? For example, i want to "get" the event when user changes view from Day to Week for example. Or is there an easy way to create a calendar lightning component from scratch ?
Short answer to your question is no. Standard calendar is a pain to do anything with.
You may have more luck with a custom component - e.g. https://github.com/Gurenax/sfdx-lwc-fullcalendarjs