If it's a monk weapon, yes. Or if you get Weapon Training with it (assuming you're high enough level to do so).
No. (Although I guarantee at least one person will post and say yes) The proficiency section of the Sohei is an addition to, not a replacement of, the normal monk proficiency section, so the part that says you can't flurry in armour still applies. The person who wrote the archetype has said as much on these boards in the past.
As long as you aren't wearing armour you can.
You cannot use a great sword using the rules from Sohei, but there is something strikingly similar. You see, in the polearms weapon group, they placed a weapon from the heavy blades group. This is called the nodachi. If you picked pole arms at 6th level, then yes, you could flurry with it
The reason the nodachi was placed in the spears group was that it likely had similar properties: it was too large for use in close quarters, it could be braced, and generally it was meant to be used against cavalry (although I think it would work equally well against dragons) Have fun with that.