Re: Solving PI vs. time to mine one Bitcoin...
If you really want to understand Bitcoin, then you really should study from the people who work on it, such as http://bitcoin.org/en/how-it-works
Question 1: Is mining Sisyphean?
I can't tell whether Bitcoin mining is Sisyphean. The story of Sisyphus is a morality tale about the futility of undermining the rules of the gods. The problem with Sisyphus was that his ultimate opponent had divine powers. Bitcoin has no gods as enemies, as far as I know. Likewise, not everybody is involved in Bitcoin for the monetary reward.
The calculations are not to slow the speed of the expansion. They do, but their purpose is to reinforce the integrity of the Bitcoin system. Bitcoins are traded from one wallet to another via transactions. Transactions are listed in blocks of transactions, in a chain of blocks going back to the first block. Zero or more transactions are bundled together in a block, along with some additional data such as the identity of the previous block, and the block is validated as being part of the block chain by having a header with the appropriate difficulty.
The hard part is finding a header with the appropriate difficulty. The difficulty is adjusted up and down depending on the speed of finding the previous several blocks, so each block takes on average 10 minutes. The Sisyphean aspects are that:
1) The more miners working on the blockchain, the less likely you are to find a block, so your mining equipment becomes less valuable. CPU mining is already worthless, and GPU mining should become unprofitable due to competition with FPGA and ASIC.
2) The reward for finding a block is a combination of the transaction fees and the new Bitcoins themselves. The number of new Bitcoins is gradually decreasing, by design, until just under 21 million Bitcoins will be generated by 2140. Assuming the Bitcoin system is still operational at that time. Miners will still have an incentive to mine because they receive the transaction fees, but that depends on Bitcoin becoming an active currency with enough volume to make the transaction fees significant.
Question #2: Could Bitcoin be used for utilitarian purposes such as SETI or Pi?
No.
It would have to be a different protocol, not Bitcoin, because Bitcoin is inextricably linked to the use of cryptographic hashes to find headers of appropriate difficulty. I guess you could make a newer version of Bitcoin that uses a more difficult hash, if SHA256 turns out to have a fatal flaw within the next hundred years, but it doesn't work for general-purpose computing.
The great thing about the cryptographic hash is that anybody can verify the headers, much more cheaply than the headers take to calculate. You could hypothetically mine using SETI, but there's no way for somebody to independently verify that the SETI-miner has done the work. They would have to download the same blocks from SETI and do the calculations again. That's a lot of work and a single point of failure.
Also, solving pi is an extremely impractical method of compression. For one thing, there is no "solution" to pi because it's infinite. For another thing, to use pi for compression, you would need to store the "solution" somewhere so you could use it for compression, which would more than wipe out any savings. Finally, because it's infinite, the number of bytes you would need for the index is also infinite, so in general you don't save any space in the transmission. I'm sure better mathematicians than I have proven it somewhere.