Can I register the name of an app before the app is finished?

There is a distinction between your goods and the app with which people may purchase those goods. Use in commerce is a prerequisite for USPTO (i.e., federal) registration. If you have not yet used your brand name or logo ("Marks") in commerce with your "self-designed items," you may still file a Section 1(b) intent-to-use application. Your Marks will be examined for register-ability and whether such registrations would create likelihoods of confusion with preexisting registrations or live applications. If everything looks ok and your applications are not opposed, you may file a Statement of Use once you are using the Marks in interstate commerce, thereafter acquiring your registrations. To answer your last question, your Section 1(b) applications can generally act as a placeholder for your Marks, though your ability to ultimately register them will depend on others' preexisting uses of similar marks for similar goods. Your ability to use the Marks in commerce does not have to depend on whether your app is live (e.g., you could sell the goods over a basic website). The nature of your questions highlight that it would be very prudent to speak with a trademark attorney in private to evaluate your proposed brand name / logo and larger branding strategy (e.g., domains, social media handles, etc).