Yes, you are correct. There is a fixed order, as ensured by the merkle root and the proof of work. Both the blocks and the transactions within each block are strictly ordered.
This order is actually necessary for validation of the transactions within. For example, you could have a transaction Y that depends on the output of transaction X in the same block. If Y came before X in the list of transactions in the block, then the block would be an invalid block.
With that said, although the order of transactions within a particular chain snapshot cannot change, the chain itself can change from time time. This happens when there is a fork in the network. Look at this image below:
Let's say you knew about block A and you thought it was the tip of the best chain. Then all of the sudden the other chain comes out, and block A is replaced by block B. The order of transactions in A might not be the same as the order in B.
It may also interest you to know that to enable Gavin's
O(1) block propagation proposal, there is a canonical ordering for transactions in a block.