Will deleting the Body Field from one Content Type also delete this field from other Content Types? When creating a new Content Type that does not need the body field, is it safe to just delete the body field from that new Content Type without affecting other Content Types?
Yeah is safe, but directly you can't delete the body field, in the page where you admin the fields there is no link for delete it.

The way to delete it is when you create a new content type in the field where you put a label for "body" just let it in blank.