Here are some links you may find interesting:
https://en.bitcoin.it/wiki/Block_hashing_algorithm
http://bitcoin.stackexchange.com/questions/148/what-exactly-is-mining
You could also read the source code of several mining clients on github if you really want to get into the details.
Are we getting our work from Satoshi?
No. The Bitminter server puts together the data explained in the links above according to the bitcoin rules for a block. It generates a lot of different variations, and your mining client uses this data to generate even more variations of a block. For some of the data there is only one valid value at a given point in time. Some data is just random. And then there are all the transactions. If we mine a transaction you made then our work data was influenced by your transaction. Can you send a "magic" transaction that would instantly cause our pool to find a block? Read on...
Can't we improve luck by changing the data we are hashing, if we have bad luck?
We are changing the data. Billions of times per second. That's what mining is.
Can we improve our luck by changing the block data we mine in a certain way?
Only if you find a shortcut/weakness in SHA-256. There's a good explanation of the algorithm at http://en.wikipedia.org/wiki/Sha-256 - see if you can find a weakness in it. Weakness meaning that you can with better than 50/50 chance determine whether a certain input will give a low hash without actually doing the calculations. Please, no superstition or hand waving. One person once told me that if the pool has bad luck I need to generate a new pool wallet to break out of the bad luck. While you can't break or weaken SHA-256 by saying random things that indicate severe mental problems, you CAN perhaps do it if you are able to untangle all those calculations involved and find a way to predict how a certain input will have certain effects on the output.
Isn't future luck influenced by past luck or other past events?
No. You are suffering from gambler's fallacy or something similar. Probability is hard for humans to understand, and misconceptions are extremely common. That's how Las Vegas was built.
Wikipedia has a lot of interesting info on how your own mind can screw you over.
https://en.wikipedia.org/wiki/Gambler%27s_fallacy
https://en.wikipedia.org/wiki/List_of_cognitive_biases
https://en.wikipedia.org/wiki/List_of_fallacies
Yes, it can.
But that's not a small thing. The way the blockchain solves the Byzantine Generals Problem is a breakthrough in computer science. This may be more important than Bitcoin itself.
To understand how impossible it is to use bitcoin mining hardware to do something useful other than mining, read my answer to this question on the bitcoin stack exchange:
http://bitcoin.stackexchange.com/questions/7236/bitcoin-mining-asics-used-for-cryptographic-application-rainbow-tables
No, that's not what we're doing. You can't even crack a simple password with a bitcoin miner (see above).
We know exactly which data is being hashed. We know the exact algorithm we are using. And we know what the purpose is. We also know it is damn near impossible to use this for anything else (see above).
If you are a little more curious you can just look all of this up. All of this information is right in front of you. It's not at all like we'll never know.
Thank you Dr, I shall have a read when I get some time.
You gotta admit though, it would make a seriously swanky crypto-vault in the Cloud. A hash that can never be broken, as it is updated so quickly by us miners, is a very nice hash...