As for your trailing /, IMHO, that's VERY bad as that is telling your visitor that your request is one level deeper in the file structure than it really is (making relative links miss the requested file by one folder level),
Then, your first mod_rewrite block is not changing the query string so there should be no impact.
Finally, the problem is that you're also making the new link's query string into a filename. How is Apache supposed to know whether it's a file or query string?
Recommendation: Get rid of the trailing / and turn events.php?name=event-name into events?name=event-name.
It's all simple logic - have a read of the tutorial linked in my signature.