Hashing Transactions for Mining
Yes, the order matters. Changing the order of bytes of data being hashed will result in a different hash. Thus changing the order of transactions for the merkle root will also result in a different merkle root. There is no requirement that miners try different orders of transactions. But they may choose to do that instead of re-selecting transactions for the block. Note that transaction order does matter (coinbase must come first, parents must come before children, etc.) so a miner cannot try all possible permutations of transactions.