I was actually involved in a computer program copyright case years ago.
Understand I am NOT an attorney, also, it's been a while...
I don't think you can copyright an app.
An app is a computer program and the computer program copyright is on the source code. Or at least you copyright the computer program by submitting the source code.
When in court, the other party was forced to provide the "source code" for their computer program.
At least one question was before the court. Who's source code was this and who wrote it.
They claimed the source code that I wrote was not used and someone else's code was used.
The source code actually had my name in it. I had hidden my name within the source code in such a way that a search wouldn't find it.
Long story short
It came down to who owned the computer program (not the idea, but the program) and if I owned it in whole or part, did they make any money from it.
I proved that I wrote the program and the source code was mine, his claim that it wan't used at all was proven false.
He didn't make any money from it, so there was no money damages awarded.
These cases usually come down to money damages, this might have changed. Proving the case was a real chore, lawyers are expensive, and people lie.
The people lie part is hard to get over, even when he was caught in a lie, it becomes an issue of how were you actually damaged by this lie.
If you think someone automatically gets into trouble for lying in court, no.
In addition, my understanding is that you automatically have a copyright when you write a program. The sending in source code and forms, just registers the copyright you already have.
Understand that things might have changed and this was a while back.
IMO, I would NOT consider a copyright registration until you have a hit, and even then I would ask a pro what advantage it would give you. My guess is that answer will be "it won't hurt, and can help protect you" without details as to how it would help you.
The place it helps you is in court, that after you have served someone. I tried to serve a drunk driver with info from the police report. Everyone involved said this person had left the US and they never expect to have contact with this person ever again. You can't get info from some sources like ins companies or police.
The bottom line in my view is that nobody cares except you and people that can charge you big $$ to "help you".
If you get a big hit, hire someone that can give you specific answers.
Otherwise, having Apple do a take down, might be your best choice. Being able to prove you were 1st and they weren't might be worth much more.
I'd be interested to see what has happened to Zynga. They are being sued by EA over some game. THIS should give many answers. I'd see exactly what EA does to protect their works.