1) From what we can tell, Kyoshi is taken. All of the territories were taken except Zaofu, and I assume it was either 1 of the 2 states from E1(along with Omashu, there is no proof that these were the two, its just my personal headcanon) or part of the 90%. I assume the other 10% was Yi and Zaofu. Just pure speculation here, no basis on fact. While it is safe to presume it is part of her empire, there is still a possibility it is not. We don't know(I personally would have liked in E1 a line up of all the states named, but that's just me) whether or not its part. The only CONFIRMED states would be Zaofu, Yi, and the rest of the land based EK. So Omashu is definatley taken. Kyoshi Island remains in doubt, and I doubt we will ever know.
2) The Kyoshi Warriors are either: serving in Kuviras military(they might have been forced when she took over, I doubt they would volunteer), or more likely, at the island, at least playing lip service to Kuvira. If Kyoshi Warriors can't stand against Azula, I doubt they could stand against Kuvira. If Kyoshi is free, their probably just hanging out on Kyoshi. Some might have volunteered to join the military of Kuvira to help reunite the empire, but no evidence of this has been shown.
NOTE:all of this is from my conjecture and deductive reasoning, I have no actual basis on sources for my guesses(while I do draw the guesses from fact).
Either conquered or joined her willingly. If the Kyoshi Warriors are still around, who knows what they were doing. Maybe Kuvira didn't want them, thinking their methods were out of date. Or maybe not, who knows?
So, out of extreme curiosity to Kuvira's take over and Kyoshi Island, I began to ponder of two maps. One of the Avatar World with a few of the major stops along the series marked down, and another of the one Kuvira kept with her on her train.
After tedious research that lasted a whole of three minutes at most, I have come to the conclusion that Kyoshi was never taken under Kuvira's control. Between the two maps, the island where Kyoshi should be on Kuvira's map does not exist. Of course, I'm 99% sure that Kyoshi Island was not obliterated or anything of the like, so it is my best guess that Kuvira either doesn't care about Kyoshi, or it has once again become forgotten
It could be that Kuvira left Kyoshi Island alone for political reasons.
Kyoshi established the island's independence from the EK several hundred years prior to the series. In that time, apparently all parties have treated it as independent, albeit irrelevant. Even the Earth Kingdom apparently acknowledged it to the point they worked together in the war. It could be that Kuvira simply treats them as another nation. It wouldn't make any sense to invade a place that she doesn't regard as territory she should control. The URN on the other hand, she did regard as her territory, so that at least makes sense.
The resources on Kyoshi island are negligible compared to the total resources of the EK. It would be expensive to invade, which would reduce returns further; and it would be a logistical nightmare for transportation and administration. Kyoshi's only value is that it's the halfway point between the mainland of the EK and the territory of the Southern Water Tribes, which don't seem to be of political interest to her.
Kyoshi Island is a minor independent state between the EK and the southern water tribe, like Finland was between Sweden and Russia for a while. The main difference being that Kyoshi Island is habitable. If Kuvira invaded Kyoshi island, the southern tribe might view this as prelude to an invasion of their territory, since its only value is as a halfway point between the two. If this happened, the Southern Water Tribe would probably go to war for their own safety, and drag other factions, such as the northern tribe, in for mutual defense. Kuvira knows this, and she's not stupid enough to risk a war with at least 2 other nations to conquer one small negligible island, which would likely have a strong resistance rendering it useless anyway.
Every nation needs a minor proxy state which can act as an intermediate when direct contact isn't viable, get around sanctions, create the illusion of cooperation, accounting fraud, etc. The US has Israel and Canada. Russia has Belarus, China has Pakistan and sort-of Hong Kong, the UK has the Netherlands, India has Nepal, etc. I think it would make sense for Kuvira to keep Kyoshi independent, purely so she has a proxy state to run things out of. She's politically savvy enough to do something like this.
Only a massive idiot would start a trans-oceanic war when they had massive insurrections in their territory, rouge provinces, and what they viewed as an occupation state in their borders. This sort of logistical stupidity is the reason Japan lost WW2. Kuvira has bigger ElephantKoi to fry.