When I select an item I want to check some fields before it will be displayed in the Field Editor and then change values on other fields.
So I need to subscribe to an event, but such event doesn't exist out of the box as I can see. Is there a way to hook to item selection action or I need to create a custom event, if so - where do I need to raise it?
Sounds like you need to create a custom validator - this blog post describes the process: https://www.habaneroconsulting.com/stories/insights/2016/creating-a-custom-field-validator-in-sitecore
In summary:
Create a new field rule (Field validators are located in /sitecore/system/Settings/Validation Rules/Field Rules/) linking to your assembly. The blog post above gives the following example of a field validator
Credit to: MICHAEL ARMSTRONG