How to add case fields to activity report?
Search - Custom Searches - Activity Search supports case id at least already, but the display template explicitly doesn't show it, although it uses it to create links.

Doesn't look like it would be difficult to modify Activity Search. You might also be able to modify the Activity CiviReport similarly. For Activity Search, it would be something like:

1) In CRM/Contact/Form/Search/Custom/ActivitySearch.php:
(a) Near the top where it defines the columns, add your columns labels as desired.
(b) In function all(), add the case-related fields to the select definition.
(c) In function from(), add a left join on civicrm_case.

2) In templates/CRM/Contact/Form/Search/Custom/ActivitySearch.tpl, display your new columns as desired.