Who runs the Bitcoin network, and how do they adjust the rate at which the coins are created?
TL;DR: The Bitcoin protocol runs the Bitcoin network and the rate at which coins are created cannot be adjusted manually, but follows a pre-defined reward schedule.
The money supply in Bitcoin is governed by the rules defined in the Bitcoin protocol. The protocol defines a reward schedule, i.e. how many Bitcoins may be created with each block, and the rules by which valid blocks may be created.
The Bitcoin protocol allows a miner that finds a valid block to create a specific amount of new Bitcoins and sent them to an address of the miner's choosing. To do so, the miner includes the mandatory coinbase transaction in the block which doesn't have an input, but only outputs the newly generated coins.
For Bitcoin, this block reward started out to be 50 BTC per block. It is halved every 210,000 blocks, so that it is currently 25 BTC per block (as we are currently at block 281,535).
Block interval and difficulty
The protocol defines that any block which doesn't fulfill the current difficulty criteria should not be accepted by the network. The difficulty is adjusted every 2016 blocks (approx. every 14 days). By the rules defined in the protocol, the new difficulty is calculated such that there would be approximately one block every ten minutes, if the amount of the work available to the network remained like it was for the previous difficulty period.
Via difficulty: https://en.bitcoin.it/wiki/Difficulty
Simply put, the network tries its best to make sure a block of 25 BTC (reward will change in the future) is found every 10 minutes.