What order does Bitcoin Core place the inputs it spends in? What order does Bitcoin core place the inputs it spends in? Is it random? Is it sorted by anything? What, if anything, can you tell about the originator of a transaction by the input ordering?
The inputs are chosen by the function SelectCoins() in CreateTransaction() and then added one by one (without shuffling) to the transaction.

The order is defined by the coin selection (CWallet::SelectCoinsMinConf) which does a random shuffle of the potential useful inputs and a stochastic approximation.

Mind also that Bitcoin-Core's coin selection could be improved and there are several issues and improve proposals ([1], [2], [3])