Does it effect your game or not? People have been telling me that it's fine to time travel backwards and you won't risk your villagers moving out even if it's a few months. Is this true? I need to get back to September from December! 

Do villagers decide to move when you TT backwards, or is it only if they say "I'm moving out on X day" that this is affected?
Trees actually do grow while TTing backwards, and flowers do spawn (saying this as someone who frequently TT's backwards). Wouldn't be surprised if the same was true for mushrooms.
Well I buried fertilizer, TTed back 2 days, and it was still there.