If you don't want them to subscribe during registration then edit your plan in CBSubs > Plans and under Workflow tab set "Allow registration" to "No". Suggest also setting "Allow Free Registered Users (without subscriptions)" to "Yes" in CBSubs > Settings; note you can also globally toggle off registration plans here with "Show subscription plans at registration". Alternatively you can make your plan non-exclusive (also treated as mandatory or required in CBSubs) so it becomes optional during registration. Aside from that I don't understand what you're asking.

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Is it possible to create a subscription-only application form in the menu separately from the free registration form?

A user needs to be registered to subscribe. Yes, you can make a menu item to view and subscribe to plans as explained below.

www.joomlapolis.com/forum/277-cb-paid-subscriptions-support/244315-about-cb-subscription?start=0#326379
www.joomlapolis.com/forum/277-cb-paid-subscriptions-support/244315-about-cb-subscription?start=0#326429

If possible, I would like to change the application format and layout.

What are you wanting to change exactly?

As an addition, when I create a module, it is blank and nothing is displayed.

It only displays if you have plans available to the viewing user. Setup your plans correctly first then worry about everything else after that. To setup your plans create them in CBSubs > Plans. Review the Workflow and Access tabs of your plans carefully as that's where most of the behavior is for controlling access and visibility of your plans.