Let's say I have deployed a smart contract game that rewards the first user that writes to contract after a point in time T. Naturally, every player will try to be that winner, causing many requests to hit the network at once. If anyone could go into as much technical detail as possible, how does the Ethereum network determine who would be the winner of this game?
There are no actual race conditions in the Ethereum network. All transactions are executed after each other and in the end there is no parallelism. You can read a bit more here: how to deal with race condition and execution times of a contract? and What is the order of operations to verify a transaction?
So basically it's the miners who decide on the actual order of the transactions in a block. Typically miners prefer transactions with a high gas price because they get to keep the gas in the blocks themselves. But there is nothing enforcing any particular order: a miner can include whatever transactions he wants in a block and it's as valid block as any other (if he manages to solve the PoW task of finding the right nonce value).
If you are indeed checking for transactions which occur after a certain time and not after a certain block then to have the biggest chance of winning a participant should basically issue his transaction at exactly the right time (probably a bit before the given time because transaction propagation takes time) with a high gas price. If the price is right, it will most likely be included in the next block and possibly even as the first transaction. But as I said there is no way to be sure of this - miners decide on the eventual order of transactions and which to include in a block.