Regarding Prince Of The Blood and Lord Nirome's punishment - How can a name be forbidden to Keshian noble children while at the same time being removed from all written documents? There would be no way to record the names the nobles couldn't use, meaning that such a law could only work for as long as human memory.
If Kesh were entirely dependent on written records, that might be true. But given Kesh's culture is based on a society of hunters, they probably depended entirely on their bards and wise men passing on knowledge through oral history from generation to generation. And given the level of bureaucracy in the Keshian Empire, odds are there IS some functionary whose job is keeping tabs of the names of the noble lines who can be told "Nobody uses this name anymore." Then he tells his apprentice that name is forbidden. And he tells his apprentice. And so on and so on until the original offender is completely forgotten in the oral history, save as a reference to something forbidden that is known only to one Lore Master.
Forgetting precisely why a name was ever banned in the first place is probably the last stage of the person being completely forgotten.
In Daughter of the Empire, why were Lanokata and Sezu part of the war on Midkemia? The Acoma were members of the Jade Eye party, while the Alliance for War consisted of the War Party and Blue Wheel Party.
I believe the Jade Eye party were part of the initial Alliance for War when the rift to Midkemia was first discovered. Later, the Blue Wheel Party would rejoin the War Party to form the second Alliance for War during the later half of The Riftwar.
In addition, what in the world was he thinking letting his house become so vulnerable? As we see with the Minwanabi and Shinzawai, there was no obligation to send the near-entirety of a House's army, nor was there an obligation for the Lord and Heir to both be present on the battlefield. He simultaneously left his house's natami protected by only fifty soldiers (not even leaving a notable commander) while in a feud with two of the greatest houses in the empire, while also making sure that he himself and his heir (soon to be sole heir, yet left without wife or concubines) were constantly simultaneously being sent into battle (knowing that they would likely fall under the command of a Minwanabi). And, apparently, he did all this for what seems to have been for zero gain. It's like he wanted his house to be destroyed.
The short version is that he gambled much and lost all.
The long version involves you considering the mindset of the Tsurani commanders early on in The Riftwar and their presumption that the people of the Kingdom of the Isles were simple barbarians who would be easily conquered. The only reason the Tsurani held their own during the early part of The Riftwar was due to superior numbers and their willingness to throw good men away to secure a victory to bolster the glory of the commanding officers and The Warlord. It took several years before the officers on the front lines like Kasumi noted the innovations that The Kingdom had (superior building techniques on their forts, cavalry and tactics designed to keep men alive rather than simple swarming) but they had no luck in getting the tradition-obsessed leadership to acknowledge this. Given that, it seems likely that The Lord of the Acoma joined the battle and took his heir with him expecting a chance to win great honor and glory for his house in what promised to be an easy victory.