Hi Adam,
It is possible to remove any content you would like. Removing the Categories or Tags from an events would require some changes to the templates. You can find instructions on how to move and modify templates on our themer’s guide.
https://theeventscalendar.com/support/documentation/events-calendar-themers-guide/
As for hurting with Google searches we cannot say that is up to you and your site design and what you are trying to appear for in searches.
Does that answer your questions?