1) What are the duties of Hakurei Reimu exactly?
Does she perverse a state of balance in Gensokyo and/or to protect human or what?
Solve incidents and do violence/exterminate to youkai.
3) Do you think youkai in general (mainly Ghost) need to eat in order to survive?
Youkai, not sure. Ghosts, probably not. It probably depends on the type of youkai.
4) What do you think Reimu strongest attack Musou Tensei do exactly? (I'm not talking in spell card battle)
Its just an attack with her throwing lots of ofudas and stuff. I imagine it isn't too different from her other attacks, except, stronger and makes it extremely difficult to hurt her.
The Barrier itself is the thing that "freed them from the spell of human imagination" so it's doubtful that they'd still be free if they left it. Although it's certainly not impossible for them to live out there, since Sado is apparently home to a large population of Tanuki. Maybe Tanuki are special in some way though, since they achieve their "purpose" by blending in with humanity.
I don't think it was talking about the Barrier that was doing it. I think it is meant to be the youkai of Gensokyo that was free from that.
@Spell Card Rules:
Spell card system is a set of rules both parties agreed to adhere to. They do not necessarily have to follow it. Youkai can indeed kill Reimu if they decided not to adhere to it.
@Hakurei Barrier: If Reimu were to be defeated, the barrier would likely be in trouble, if not disappear. Here is the source, from Perfect Momento in Strict Sense, Reimu's profile:
The Hakurei shrine maidens control the Great Hakurei Boundary and protect Gensokyo.
If the Great Boundary were destroyed, either from inside or out, Gensokyo would likely not fare well.
Gensokyo relies on the Hakurei Shrine and its shrine maidens to continue to exist as it does today.
As a result, no youkai oppose them.
@Youkai sages: I don't think there are that many youkai sages, but I believe there are more than one. There are several people I can think of that can fall into the category of youkai sages. Tewi, Aya, Yuuka, and depending on your view, Keine.