Ok, I managed to solve this. Here's what I needed to do, in case someone else runs into these issues:
1. Enabling the Bootstrap CSS fixed the layout issues (Configuration -> Component -> Load Bootstrap CSS). The previous version worked ok without it.
2. Submission button appeared once I set the toolbar position to "Tob and bottom" (Configuration -> Event Editing -> Toolbar position). It was set to "Top" and, again, worked fine in the previous version.
3. The image button was actually part of the editor window, so I removed it by going to Extensions -> Plugins -> Button - Image (for editors-xtd) and set the permission to "Special".
I still need to do some shivving to remove the "Cancel" button and rename the "Save & close" to "Submit", but at least it's working again.
Just a heads up: When I submitted a multi-category event and later viewed it in the backend, it gave me two warnings:
Notice: Undefined offset: 5 in /[...]/components/com_jevents/libraries/jeventshtml.php on line 302
Notice: Trying to get property 'value' of non-object in /[...]/components/com_jevents/libraries/jeventshtml.php on line 302
The relevant part of code is this:
if ($options[$o]->value == $catid[$c])
{
$options[ - (count($catid) - $c)] = $options[$o];
unset($options[$o]);
break;
}