The TfL mapping guys have indeed jumped the gun. Though it's hard to prove a negative in this case, the Heathrow Express website is correct. The most official source you'll probably get, the Heathrow Express Conditions of Carriage, state unequivocally that Oyster is invalid.
However, Oyster is expected to be implemented imminently (I believe estimates were previously some time in December, and even if it's slipped I don't imagine it's that far away), so it's likely that TfL made the map with the consideration that it'll end up being true for most of the map's validity period. Maps are only put out twice a year, so this is not the first time this has been done (a change made a little early so the map will be correct for most of the period) and it probably won't be the last either.
As to the second, I can only speculate, but I'd strongly suspect the differing gatelines at London Paddington station have something to do with it - this is also how it's implemented with the Gatwick Express at Victoria. Heathrow Express trains tend to leave from platforms 6 and 7, whereas TfL Rail leave from the high-numbered platforms (or eventually the underground Crossrail platforms). This can change during disruption, but I don't know whether someone will take the revenue loss, whether the barriers will be reconfigured during disruption, or whether the onboard staff will have Oyster scanners which influence which fare is charged.