You'll get a new set of quests (visiting neighbors/community kingdoms and defeating fang beasts) every 23 hours. Completing those will give you midnight mushrooms to chop.
No, they don't accumulate. But there's no limit to the energy amount you have so say you collect energy, not use them, or use less than what you collect, voila you can accumulate quite a bit of energy. The largest energy stored known to date: 100,000 energy. I'm sure you can store more and to be honest I'm not sure if there's a limit. The only person who can test that limit is Puffhausen but I don't know if he plans to play anymore. Haven't seen him in a while. Does anybody know where Puffy is?