The Related Record Component is linked to quick actions so you can control the fields that display/layout through the quick action it's tied to.
In your specific example, there does seem to be standard actions that aren't customizable when you use the default when dragging a related record on a case. These are called "Update Case" and "New Case". It's mentioned in this question.
You'd want to create your own update a record quick action type on your case object. On that quick action, you can dictate the layout and predefined fields.
When you drag that component, you can select your custom quick actions which will then display your configured layout.
In terms of limits, same rules apply as quick actions. You can technically drag all the fields available (no imposed limit), but this recommendation displays once you add more than 8 fields
Adding too many fields to an action layout can impact user efficiency. We recommend a maximum of 8 fields. To reduce the number of fields, you can set predefined values for fields that need a value, but won't often be edited. You can safely remove those fields from the layout. Set predefined values from the action detail page.