Am I correct in my understanding that:
a) There is a single block chain for the whole world at all times
a) There is a single block chain for the whole world at all times
No. The end of the chain can, and does, fork, but the forks are short and the network pretty quickly decides on the One True Chain.
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b) The block chain contains a record of every bitcoin transaction that has ever taken place
Yes.
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c) The entire block chain must be downloaded in order for a client to use bitcoin
Short answer: no.Longer answer: it is complicated, and what you need depends on whether or not you're trying to generate new blocks. To keep it simple, the original client downloads everything.