I would like to access the search engine via https://. Does Alkaline support SSL? Can the Apache proxy mechanism be configured to filter requests and forward them to Alkaline?

Alkaline does not have a native support for SSL. You can setup Apache to forward requests to Alkaline thus exposing an SSL front-end, so search can only be performed via SSL.

Modify httpd.conf to enable proxy: 
<IfModule mod_proxy.c>
  ProxyRequests On
  ProxyPass /search http://www.search-srv.com:9999/
  NoCache *
</IfModule>

The VirtualHost directive should contain the same thing, here is an example: 
<VirtualHost www.search-srv.com:443>
  ...
  ProxyPass /search http://www.search-srv.com:9999
</VirtualHost>

Add a directive for all requests to go through SSL: 
<VirtualHost www.search-srv.com>
  ...
  Redirect permanent / https://www.search-srv.com/
  RedirectMatch permanent (.*) https://www.search-srv.com/$1
</VirtualHost>

Add <!--SET SEARCH-BASE-HREF--https://www.search-srv.com/search--> to the search template.

This has been implemented, courtesy of InterNAP Network Services (http://www.internap.com/).