Question: Do the REST API and all other XML/AJAX function work with self signed certificates?
Using my test server I get complaints about the self-signed certificate. If I use HTTP only everything works.

From the Settings/Status page:

REST API
Failed 
https://vhost1.site/api/v6/collections
This API doesn't work properly on this server. You should probably update your .htaccess file to the latest version and check the file permissions of this file. Error: SSL certificate problem: self signed certificate (#60); Status code: 0
XML-RPC
Failed 
https://vhost1.site/xmlrpc.php
This API doesn't work properly on this server. Error: CURL error: SSL certificate problem: self signed certificate
AJAX anon_async.php
Failed 
https://vhost1.site/htsrv/anon_async.php?action=test_api
This API doesn't work properly on this server. Error: SSL certificate problem: self signed certificate (#60); Status code: 0
AJAX async.php
Failed 
https://vhost1.site/htsrv/async.php?action=test_api
This API doesn't work properly on this server. Error: SSL certificate problem: self signed certificate (#60); Status code: 0
AJAX action.php
Failed 
https://vhost1.site/htsrv/action.php?mname=test_api
This API doesn't work properly on this server. Error: SSL certificate problem: self signed certificate (#60); Status code: 0
AJAX call_plugin.php
Failed 
https://vhost1.site/htsrv/call_plugin.php?plugin_ID=-1&method=test_api
This API doesn't work properly on this server. Error: SSL certificate problem: self signed certificate (#60); Status code: 0