I'm curious when bitcoin miners solve those complex algoritms does that mean a new block is added to the block chain that confirms the transactions then that leads to a reward of bitcoin for the miner OR do miners both confirm the transactions on a blockchain or ready there AND solve the math algorithms to be rewarded with bitcoin?

So JP Morgan Bank has a digital coin. How does it work out when a business wants to create its own digital coin? Does this mean they buy a data center and try to mine coins or what? I understand something like ethereum as it is for smart contracts but having problems understanding altcoin.