Hello. Check this file (based on version 2.5. :
/components/com_dtregister/dtregister.php
Around line 2984 you should find the start of a <th> tag that includes the header for the date column shown as 'DT_DATE'. In this table structure, you would need to remove that column starting with this line from <th> to </th>
Then the actual data for the date is pulled in around line 3306 where you will see the start of a <td> table cell. You should see some clues that you're in the right place like "get_date", $start_date, etc. Comment out or remove this code from the start of the <td> to the closing </td> which about 7 lines later.
casey, removing the category has been posted in the forum a couple times before so a search should locate that. The new 2.6 version has new parameters built in allowing you to "Hide" the search and/or the category filter. That is almost done!