Could any standard fee transaction be double spent by using a higher fee? I realize that if somebody makes a transaction without any fees and there have not yet been any confirmations, then the transaction could be double spent by just spending the same inputs to a different address, but with a standard fee. 

My question is, if somebody sends a transaction with a standard fee, and there have yet to be any confirmations, could that transaction be just as easily overridden by rewriting the transaction, but with a higher fee? If so, what would be the implications for a party that accepts zero confirmation transactions such as BitPay?
The original design assumes a common behavior of processing the incoming transaction on FIFO basis. That means the first transaction received by a miner wins and all conflicting transactions received after it will be ignored (no matter if the first one is already confirmed or not).

While this standard behavior is beneficial for the Network as a whole, some "smart" miners might employ a bit different approach: when building a block, always take the transaction with the highest fee, no matter the order in which it was received.

It might seem this is just a marginal misuse of the protocol and there won't be enough rogue miners doing this, so it doesn't matter. Well, unfortunately, there already is (at least) one mining pool that tries to build its business exploiting this idea. Should it prevail, services accepting zero-confirmation transactions will need to reconsider their policy.