Please do not create duplicate threads. Have linked your previous thread to this one and locked your previous thread. Please understand your PM was received on Saturday, I am not here on Weekends except for a minimal amount of time to address emails. Onto your issue.
You're using a CB 1.4 on a J1.6 install, please upgrade and remain upgraded. There are known issues with CBs core regarding J1.6 and J1.7 critical to your installs success; whether this applies to CB Conditionals or not is irrelevant.
The reason the registration is not behaving properly is because you've a jQuery conflict. The below fatal JS error is preventing CBs validation code from operating properly.
Code:
jQuery.validator is undefined
index.php/community/login-register
Line 34
The above happens with or without CB Conditional and is clearly not the cause. CB Conditional is tested as working perfectly fine. The problem is the conflict preventing it from working as it's 90% jQuery (JS) driven.
Upon further review switching to Default Joomla template resolves the issue. This is due to your Template loading jQuery. This normally isn't a problem, but your template is loading 1.6, which changes how many selectors operator and results in many if not all jQuery calls written for <6.0 failing. Please contact your template developer and instruct them to run jQuery in a noConflict state (routing $ to their own custom selector) or checking if jQuery is already loaded before loading it.