I think there are some "devil in the details" that you left out, so I'm not sure I can give a complete answer, but here are my thoughts:
FooCoins aren't mined via processing, they can only be bought by trading physical items and standard currencies
So you're going to start the pool with a fixed number of FooCoins. Who starts with the initial FooCoins? More importantly, who decides who gets them? You? Sounds a lot like a centralized bank to me...
Paying newly-minted bitcoin to miners serves two purposes:
Introduce coins into the market in a merit-based way.
Miners verify the integrity (trustworthiness) of the block chain. Bitcoin would not work without a very large number of miners spending a huge amount of real-world money on electricity to do the mining. Unless the reward is greater than the cost, nobody will mine and the crypto-currency will fall over.
Let's take a brief stroll into how a bitcoin transaction and bitcoin mining works:
Some person wants to send money from their wallet to somebody else's wallet, so they broadcast a message to the world announcing their intent to do this. (the message is signed by their wallet's private key).
Miners collect these announcements that are flying around and roughly once per 10 minutes, they hash them into a block. The decentralized and proof-of-work nature of bitcoin comes from the fact that miner's can't just publish a block, they need to play with inserting random values into the block until the SHA-256 hash of the block has at least N leading zeros (N is called the work-factor, and get adjusted based on the number of miners in the system). The first miner to find a successful combination of random numbers gets to publish that block, and gets rewarded with a bitcoin (plus any tips in the transactions).
So basically, we have a non centralized chain verification based on hash and random number puzzles. A signed intent to transfer message bitcoin is sent to all miners and they all compete to solve a problem using that message. Once a miner successfully solves that block, a new block starts.
This has a striking resemblance to your suggestion:
they have a non centralized stack verification based on puzzle solving. An encrypted code is sent to all the other phones in 3G and they all compete to solve a problem using that code, and another challenge starts.
I think you just invented bitcoin.
An to your actual questions:
Q: Can computation expense still be used to verify accounts? The FooPhones all have the same processing power and they randomly can win the verification task.
At the surface, "puzzle solving" still sounds like "proof-of-work" to me.
I don't think you've told us in enough detail how the verification works. What exactly is that "code" they send each other? What's the mechanism for transferring part of a FooCoin from one FooPhone to another? What puzzle are the "miners" solving?
Q: Can FooCoin OS on mobiles be hacked by fast PC's that use false ID's?
What ID's? This is the first time you've mentioned ID's in your question. How are ID's related to FooCoin?
But in general, it seems like you're relying on the "closed-source-ness" of the FooCoinOS as your protection. 1) You can hardly call this "non centralized" if a small number of developers controls the code, and 2) if FooCoins get valuable enough, the source code will get leaked, either through hacking the servers where the source code lives; reverse-engineering a FooPhone; or by one of the devs getting greedy and selling the source code, at which point you'll have server farms competing with your FooPhones to solve the puzzles.
