If mining bitcoins became unprofitable and people stopped doing it en mass, then who would verify transactions? Would the system cease functioning correctly?
My understanding is that Bitcoin production difficulty is inversely proportional to the amount of power being spent on mining. If the price of BTC drops so much that it's not worth it to mine and some people stop, the difficulty will drop and it will thus be more worth it for the remaining miners.
Rational miners would indeed stop mining, but fanboys would continue mining at low difficulty. (Keep in mind that some people are using "free" electricity, so their marginal cost is basically zero.) Then someone might do a 51% attack just for the lulz.
But I wonder if there is some kind of minimum amount of mining required to verify X number of transactions. If there are many transactions, but few miners, then would transactions begin taking too long to become verified?
Yes, there is a byte limit to the size of a block. If blocks become infrequent enough due to decreased mining power, it's conceivable that a backlog of transactions would build up. At the next difficulty adjustment, though, speed would pick back up again.
