For your two queries:-
one cannot customize this component and therefore cannot add custom fields:-
There is the option to add fields to event layout.
it is possible by editing the Global Action Layout, but this only changed the Global Action for Classic UI.
Editing Global action will reflect wherever it is used.
When you want to add a custom field in the new event tab inside activities on lightning record detail page layout, First check the button, link and action section under object manager of an object if there is any new event quick action or not. If there is not, then the new event action is a global action.
In order to add an extra field on global action layout, you can go to setup and type global action and select and open the new event action and edit the page layout. Add the desired field to the new event global action layout.
Note:- Any change in Global action will reflect everywhere it is being
used. If it affects your org, My suggestion would be to go and create
one object-specific quick action of type create a record and select
target object as the event. Add the desired field on the layout of the
Using Global action on the layout and having record types for the event
and task in your org needs one extra activity. You need to associate a
record type for global action in order to show those actions under the
activity tab on the record page in lightning. If you do not, even
after adding the global action in salesforce mobile and lightning
action section on the page layout, the action will not appear in
One good solution if you want to use the related list to create New event and the task is:-
- add related list single on the layout by editing the page using edit page option appears after clicking the setup icon
select the object Open activities and save it. Do not forget to activate it.
Once done, In your related list, You can see the open activities related list as you were seeing it in Classic. Even You can add this one inside Activity publisher tab on the lightning record detail page. The field displayed here are controlled by normal page layout for task and event. Refer How can I customize my tasks and events page layouts? for more information to edit the page layout of an event and task.
The advantage of using related list is the button here will give you
an option to select the record type which will never possible through
an action. Action always needs to associate with a record type, hence lose the ability.
This image tells you about related list single:-
This image tells how you can edit the event layout when using related list single:-