Conditional fields module - Required fields that are hidden are still required to be filled
All fields are set to be required in their field settings.


This is your mistake.  If you set the field to be required in the field settings, it will be required whether it is invisible or not.  So you need to uncheck this setting unless the relevant field will always have a value.

Instead, you should trigger the "make required" action provided by the Conditional Fields module whenever showing the relevant field for input.  Unfortunately, the last time I tried this in Conditional Fields, it did not work due to a bug.

Alternatives

Field Conditional State is a stable module that works similarly to Conditional Fields.  You can use this to properly set required fields.

Or, if you prefer a code-based solution, look into the #states API.