https issue (https shortcut link is not working)

I got great help from the following links and almost made all sites https enabled. But my effort for 3 nights is not going anywhere. Any help would be greatly appreciated.    

http://community.elgg.org/discussion/view/576874/how-to-add-ssl-to-my-elgg-site

http://community.elgg.org/discussion/view/1442318/https-works-fine-but-homepage-graphics-are-gone

My problem is that the simplified menu with https is not working. 

For example, https://mysite.com/faq gives the following error:

Not Found

The requested URL /faq was not found on this server.


Apache/2.2.22 (Ubuntu) Server at mysite.com Port 443
 
 
However, if I change https://mysite.com/faq to http://mysite.com/faq, it redirects to https://mysite.com/engine/handlers/page_handler.php?handler=faq  
And it works! (Actually not completely it has some broken graphics and formats.)
I'd like to figure out why https is not working for the simplified link.
So far what I did are as follows (using Apache2, 3rd party SSL certificate and elgg 1.8.14):

1- Changed 'Site URL' to https://MYSITE.com/ in admin's 'Settings'->'Advanced settings'

2- Enabled 'Enable HTTPS logins' in admin's 'Settings'->'Advanced settings'

3- in .htaccess file added at the top:

RewriteEngine On
RewriteCond %{SERVER_PORT} 80
RewriteRule ^(.*)$ https://www.MYSITE/$1 [R,L]

I tried a few different version of .htaccess from the links above, but didn't make a difference.   

Again, thank you for your help.