Line 3 is missing an "&" between ai1ec and tag_id. If that was a typo, try it with the "&":
post_type=ai1ec_event&post_status=publish&posts_per_page=6&ai1ec&tag_id=2368
Quotes will cause problems, so it makes sense that the last two would be problematic.
If you want to go another route, I would recommend using the standard WP_Query, info found here:
https://codex.wordpress.org/Class_Reference/WP_Query
Your $args would look something like this:
$args = array(
'post_type' => 'ai1ec_event',
'post_status' => 'publish',
'posts_per_page' => 6,
'tag_id' => 2368,
);
Finding the start date I think would involve creating a meta query for that post type.
After searching a bit online, I found this bit of code that could be another option you could use:
Hope this helps.