Do we get exact bonuses for stuff?
How much authority do we have, exactly?
What's our starting relationship with the Emperor/Emperor's Son?
What's our army size(s) and composition(s)?
What's an Arbitrated War, exactly?
Do we start with intelligence networks?
Is the Emperor always being male and having only one son your own invention or part of the pre-existing setting?
How sexist is the culture here?
Does having multiple schools make you less effective at any one of them individually?
How exhausting, exactly, is animating the constructs and/or giving them a soul? How long would it take and how many would I be able to do per day? And how many would I be able to start with?
How much are Lords of War likely to get involved in each other's affairs?
1: Yes. However you won't know them. Just to give it some realism.
2: Lords of War have complete authority over the armed forces of the Empire within their area of influence. And even outside of their areas of influence, their order must be followed by any soldier of the Empire as long as they don't go against direct orders from that area's Lord of War or the Emperor. You also have some limited authority over knight orders and the armies of the different Principalties, being able to request their help as necessary.
3: At the start, the Emperor trusts all of you (you wouldn't be Lords of War if he didn't). It is not unlikely that you might have fought alongside him against rebels or some supernatural threat. As for his son, he is still a teenager, but probably you have seen him training alongside the members of the Order of Heaven.
4: I'm working on the exact composition of the armies, but for numbers, it's going to be about 200000 men for each Lord of War. Those are the troops under your direct control, not counting the armies from the different Principalties and such that you might call for help.
5: Arbitrated Wars are fought when two Principalties have a problem between them and all diplomatic attemps have failed. The Arbiters set a number of objectives for the two Principalties (usually fortresses from the other Principalty). Attacks to the civilian population are strictly forbbiden. During the conflict, the armies of the area's Lord of War and his Arbiter try to make sure that no side tries to go against the rules. The Princpalty that takes all the objectives is declared the winner. This method is used to work out tensions between Principalties while making sure that damage to structures and the civilian population are minimal.
6: Yes you do, although it will be minimal.
7: Part of the pre-existing setting.
8: It depends on the Principalty. In general, men have more importance than women, but things like female soldiers are relatively common, and Imperial law gives the same rights to men and women. However women in positions of high power are extremely rare outside of the Order of Heaven, the Inquisition and the island of Daphne.
9: Yes it does. After all, it's easier to be an expert in a field than to be one in several fields. It takes a lot of time to learn new spells, specially the most complicated ones.
10: It depends on how powerful you want them to be. Creating one powerful enough to 1v1 an Arbiter would make you have to rest for the whole day, while one equivalent to twenty to thirty soldiers would only require you to rest if you created several. Giving them a soul will take most of your energies, regarless of the power of the construct. The spells themselves take only a few minutes each. Taking as an example the construct=20/30 soldiers, you could start with 30 of them. I realize it doesn't sound like much, but their psychological value is quite high. One thing to take into account here is that, during the game, you might develop/discover/learn rituals that could help you with your constructs, giving them their own source of power so that they don't drain yours, increasing their effectivity or maybe something complely different.
11: In theory, they should only get involved in each other's affairs when it concerns the borders between their areas of influence. However they are encouraged to watch over each other, both to make sure that they don't abuse the trust the Emperor has put in them and to be able to act should a situation get out of control without having to wait for a call for aid (which means that, in the past, many Lords of War have had troops secretly deployed outside their areas of influence)