Question: SPOILERS: When Sonic and Shadow land on the moon, they have a heart-to-heart conversation and appear to be breathing. But there's no air on the moon. Are we to assume that the Chaos Emeralds (which have given them both powers) are giving them the ability to breathe and speak in the environment?
Question: How did Stone see Robotnik's message at the end? It appeared to be displayed on jumbotron-like devices in big city squares, but Stone was in London and there didn't appear to be any giant screens, or even small ones, anywhere near him.
Answer: Per a quick Google search, it appears there are several locations in London where there are giant screens. He likely was running around the city in a panic (there is a doomsday weapon threatening to destroy mankind, after all) and managed to stumble onto one of the locations.
Answer: They still have the Chaos Emerald power at that point because, immediately after, they turn gold and fly away.
Honestly, that's probably the best answer.
TedStixon