Do I get access to the raw data that was tracked, such as clicks and mouse moves?

Yes, you get access to all data that is stored in your MySQL database: the data is stored in the matomo_log_hsr* tables.

You can also access the data via the API. let’s say you want to export a specific recording that can be found at the following https://matomo.yoursite.example/index.php?module=HeatmapSessionRecording&action=replayRecording&idSite=4&idLogHsr=67&idSiteHsr=2

For querying the raw data you need the idSite and idSiteHsr of the site and the idLogHsr of this recording.

When you now provide them to the HeatmapSessionRecording.getRecordedSession API, you get the full raw data of this visit (every mouse movement, scroll, etc.)

https://matomo.yoursite.example/index.php?module=API&method=HeatmapSessionRecording.getRecordedSession&format=JSON&token_auth=[YOURTOKEN]&idSite=4&idSiteHsr=2&idLogHsr=67

If you now want to get the data of multiple visits, you can use the getSessionRecordings and getRecordedSessions to get the idLogHsr in an automated way.

More details about the Heatmap & Session Recording API can be found here.