What happens if I upload an incomplete app on the App Store? I am in the process of creating an iPhone app. Someone, very recently, stole all my app files and I'm afraid if that person will submit my app before I do.

Can I upload my app, without the major features and, maybe, a little unstable, to the App Store? So, even if it gets rejected, I'd have the app name and design registered to me, and I can upload the full iPhone app, later, when it's ready. I just don't want my weeks of effort to be used by someone else.

Also, could my rejection have any impact on the future submissions?
You can upload the app without worry. 

If your app got rejected,it won't affect the future submissions and you'll keep the app name with yourself. Your app name won't be taken by any other person.

There is a feature in the itunes-connect to set the release date. Even if your app got approved, it won't be released to public.

if there are major tasks to be completed in the application, it is better to create a new application and make the app-status to waiting for uploading of binary. you'll have 6 months time for uploading binary(setup) after creating the app in the itunes-connect.