That bitcoins are infinitely divisible is one of the ways in which problems associated with deflation are mitigated once the 21M bitcoin cap is reached.
Divisibility doesn't really help with the overall money supply however. There's also the loss of currency that occurs from loss of Bitcoin wallets.
Even without deflation I think there would still be problems with high variance of BTC just from the psychological aspects of the currency. With central banks you can at least say that there are persons whose job it is to watch currency exchange rates and take one or more of many available actions to curb both inflation and deflation. As far as I can see you just don't have that with BTC (and by design). You can't increase the money supply, any action to decrease the money supply is permanent, and how can you change interest rates for loans when all of the money is already issued?
I just can't imagine BTC being useful for transactions in general (especially important stuff like payroll) when the inherent variance in value seems to require systems akin to what a high-frequency trader would use. It might be useful as a commodity for similar reasons that gold is (but with the improvements that come from its high divisibility).
Maybe this is a stupid question, but why is the mining needed at all? If we can keep subdividing them, shouldn't 1 bitcoin theoretically be sufficient?
"Mining" is really accounting. Miners add new blocks to the blockchain which provides for verification of transactions and the value of individual addresses. So providing a reward to mining (initially) was a means of providing a reward for performing an essential function for the bitcoin economy. Now that more bitcoins are in the wild it's possible to collect transaction fees, the second method by which miners can profit from their activity. Eventually the transaction fees will be more than the mining reward and the theory is that miners will remain in business in order to collect those fees.
Ah interesting, that makes sense. Can miners set their own transaction fees? And related if I want to make a bitcoin transaction can I (or my software) choose which miners to use to verify those transactions? Or are those fees determined by the protocol itself?
My understanding is that when conducting bitcoin transactions you can establish the fee you're willing to pay, the miners can optionally prioritize based on those fees. Theoretically it's possible that you'd have a transaction never verified or verified after a great period of time in the blockchain because your fee was set too low.