1. Emperor was known as divinity, to be from god himself. The shogun was the military, often dictator, power. Correct.
2. This is not known to me in any JSA I have researched, including long time students and 2 sensei I have spoken with. Depending what race, for example the average Japanese height is 5' 4", which would mean a katana of around 26" would be recommended by sensei. In ancient times it may have been closer to 28"~. Closer to 6 feet, you are talking about 29". It depends on the size of the individual, or the swordsmanship but rarely do you see anything over 30", let alone 34" thats nodachi territory, and we all know how impractical they are.
The average size if I recall correctly, today in events like Tai Kai is 29"~.
The only schools I can think of that would use 34" would be for almost for an nodachi (starting at 35"~), which were large swords for taking down cavalry or castle siege defense. There's maybe 1-2 old schools (kage-ryu is one) left not sure. Not very efficient or practical as a nagamaki would serve better for the purpose of the nodachi.
It takes a lot of skill to wield longer swords, this is why the typical size for katana is 26" 29" depending on height/body, I find it a bit difficult to wield a odachi, or cut tatami omote with it. I prefer katana, or tanto. Reaching correct hasuji (angle) becomes difficult with the longer the blade is.