Regarding the armsmen, Bujold retcons herself a couple times on them. It is explicit later on (WG?) that armsmen have to be from that district... but Bothari's not, and that never comes up when it should otherwise in WA and Mirror Dance. There's no indication that anyone other than a count can have armsmen; they're the leftovers from the Count's private armies, their armed support is reduced just to 20. There's a fan theory that some put forward that the Count gets 20 and the heir gets 20 also, but that's not really anywhere. So Miles is basically swearing armsmen on behalf of his father.

It's only legal if there are places in the score to do it, and it's possible that there are. We don't ever get a full list of armsmen. It also could be that Miles is banking on the fact that he's unlikely to get caught, that it doesn't completely count. He's not thinking about VorL's law at all.

But it's just not explained.