Please PM backend login credentials (Super Administrator permissions) and will take a look.

Have reviewed your installation and "System - ARTIO JoomSEF" was crippling frontend by querying the plugins database table, which does not exist in J1.6; so likely not J1.6 compatible. I've disabled said plugin and restored frontend access. The backend template "rt_missioncontrol_j16" locked out backend by querying a database table that did not exist. I've disabled said template and restored backend access.

Once done have reviewed frontend and you had the CB "default" template disabled, this template must always be enabled. You also had your CB template of choice "funlite" set to "Registered" access. Users not-registred can't access the template and thus can't access registration. Have fixed both and registration now displays as appropriately.

However, after reviewing registration all core fields such as username, password, email were not present. This was due to the tab they are on (Contact Info) was set to "All Registered Users". This prevents access at registration, because the viewing user is not yet registered. Have corrected this and set tab permissions to "Everybody".

However, during fixing of that found Save and Cancel buttons failed to function. This is due to a fatal JS error brought on by your Joomla WYSIWYG editor of choice. I've now set this to TinyMCE and disabled the 3rd party editor (JCE, maybe not up to date or using beta?) resolving the fatal JS errors.

On further review have also noticed in Plugin Management that you've set CB Core to registered, which means no field in existence except 3rd party perhaps will function for public users; this includes registration. I've set this back to the appropriate "Public" resulting in registration fields now properly being visible.

Please review all of the above carefully to understand why things happened the way they did (primarily permissions related). Please also confirm registration is now functional.

So, did I waste my money? I am unable to use either of those components with CB now...

More or less, yes; neither function appropriately on J1.6.

As far as I am aware JCE for J1.6 is free of charge. The link to JCE 2.0 is as follows, which is rated for J1.6 (haven't tested!).

www.joomlacontenteditor.net/downloads/editor/joomla-16

Please also note you're using J1.6.3 while J1.6.4 has just released; this too could be a factor and may want to consider upgrading. Please note I've yet to test CB on J1.6.4 so no idea if new bugs have been introduced.

I've no experience with Artio so not much to advise there, sorry.