1. It should be ok now.
2. I believe something else is interfeering. I did some testing to measure the site performance. An empty ajax request takes 4-6 seconds, which is extremely high. It means, that even before the search can start it must wait 4-6 seconds. This delay is most likely (but not 100%) caused by one of the plugins, but unfortunately I'm unable to tell which one it is.
If the delay is not there, then the suggestions would appear in 2-4 seconds tops.
You can also feel this slow loading if you for example try to open the plugins page: http://www.mission-locale.com/wp-admin/plugins.php
It takes me like 10 seconds to get there, which is too high, it should be less. Caching, seo and similar plugins can cause this.
Update: I tried to deactivate temporary the WPRocket plugin and the response time went under 4 seconds! Perhaps that's what caused it? Your site felt faster as well!Best,
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