How to theme webform's Custom Composite element? Is there any chance I can move specifics sub-element to Custom Composite? Either, I couldn't find any way how to use flexbox inside in a custom composite.  

For example, I wish to have;

[First name] [Last name]
[Yes, I am coming or no, coming] [allergies list]


Rather than all in a row:

[First name] [Last name] [Yes, I am coming or no, coming] [allergies list]


Can I also use conditional logic in the custom composite element such as if people click "Yes, I am coming" means allergies list will be visible?
You can add classes like you do for any other element. The only difference is that in the case of custom composite elements, there are no fields for setting them directly. Instead, you can go to your respective nested element of Custom composite and set your classes like

'#attributes':
  class: 
    - trip-departure-date


inside the custom property and theme accordingly. You can also use bootstrap class to change the layout.