1. Do I get a higher score if I DON'T let idols do the festival by themselves?
Ideally if you're good at performing a song and know how to manage your appeals, bursts, and memories... then absolutely you will. Making your idols work by themselves fatigues them, and you risk negative effects by using it too much. If it continues they'll eventually take a week off by themselves.
3. For the Lessons, Do I get a higher score if I DON'T let the idols do it themselves?
Even if you tell the idols to work as hard as possible, they will never come out with a Perfect + Bonus. At most they'll get to the very end of Perfect. So yes, if you can reliably get to the "Bonus" status in any lesson it's better for the Producer to manage the lessons. Telling them to handle it themselves also causes fatigue.
There are ways of circumventing fatigue to some degree. I believe if you get them to the point they will take a week off, but the next playable week is a story-related week, they won't take the week off. I've never tried it, so I'd experiment a bit to see what works before relying on it too much.
4. I want to get all the scenes with my leader Chihaya, I am worried that if I make it to the Top 20 too soon, I might miss those scenes and decision that happen if you are in the Top 50.
Daverost is absolutely on the money.
If you're worried about Promotions, then this is a possible outcome. I wouldn't worry about activating
all of Chihaya's scenes from the get go. There will be chances in later plays where she can be selected for Promotions, even if she's in a support role. If you concentrate solely on Chihaya the rest of your unit will quickly start to feel alienated.
That or you could simply reset after doing Promotions and select whatever else is available! It won't count in the long run, but if the aim is to see... that'll do the trick.
If you're worried about non-story After Communications, there's a real possibility you may not see all of them in a particular play through. If they do appear, they may be centered on your support instead. I won't say they're random, but I have no real idea how they come about and what the triggers are. It wasn't until recently that my leader had all of theirs occur in a single game (all three in a row), and my support only had one each the rest of the game.
I suppose it's their way of encouraging using the idols repeatedly in your units, be they your leader or support.