Mermaids are pretty much only relevant in the water, and considering that would be the downside of two (three if cold water) Youkai, and 3 others are not likely to go swimming in the first place, they are definitely balanced.
Water =/= weakness to fire! (Grease fires don't go out with water) Rather fire goes out when cooled or the source is eliminated (or oxygen removed). Considering their body temperature wont change with normal temperature water, nor is their oxygen removed, water is not a solid weakness. So I think it would be safe to say that their weakness would be Snow people (and vice-versa). Maybe even Crow Tengus since they can remove their source. Note that fire>anything with flesh, fur, or clothes.
Humans should probably have the same intellectual advantage as Mages, or in other words resistance to mind control abilities (clutch resistance, it isn't as passive as for a Mage of course.). They are weak to all, even themselves. (Batman is human!)
Nekos are weak to spray bottles, free fish, and swimming waters. I'd say they are both weak and strong (depending on setting) against mermaids.
Vampires are weak to pure water, though small amount wouldn't paralyze them, simply burn or sting them. I think the only time they would be fully handicapped by water would be if they were submerged, drenched, or hosed down by water. (A vampire could easily avoid large water sources, and likely run away before getting hosed down.) I'd say their hard counters are Mages with water spells, Yasha with some kind of aquarius summon, and a fire truck.
Their soft counters would be: A mermaid who catches one off guard, A succubus who manages to fly one into a body of water, and Werewolves who are evenly matched and between half moons.
Snowpeople: Not much to say about them, their powers (in my mind) could range from ice spikes from the ground, ice barriers, ice weapons, to just ice claws and cold resistance. They are basically human in their natural strength and durability, with their only advantage being the above. They are obviously countered by Youkou, (or evenly matched depending on the amount of chemistry we are going to use), Vampires (unrivaled strength and no easy way to get water, Snowpeople need to have the opening attack and location advantage to stand a chance.) and Werewolves (between crescent moons)
Werewolves are very situational, half and half at night. Even then I think they only get stronger by moon cycles, strongest at a full, normal on a a new. Even so, their counters are Vampires, Youkou, and a lucky human with a silver bullet.
Lycans having psychic powers means that even if it causes self harm, you could still potentially eliminate your target before you eliminate yourself. In a 1v1 Lycans have a large advantage, powerful burst potential, and plenty of willpower to last. Though, because of this I believe they are balanced by the fact that they lack the ability to fight multiple people at once, save for being much better physically than them.
I want to say their weakness is a specific counter, but they could easily 1 on 1 any Youkai, given it is a quick fight. Instead I will say that their weakness is anything that groups together, or anything that can resist their abilities long enough for them to handicap themselves.
Yasha: No comment, summoning is great, but getting caught without a summon is fatal. (Play FFX and try to keep Yuna in your party at all times without her dying) Werewolves are a grand counter, considering that they can use their speed to swipe the Yasha's coin. Vampires are very likely to be able to destroy any summon that is brought out. Lycans can just mental burst them as they are just human. A succubus could also be a viable counter with their attraction.
Crow: The potential for godlike is high on this Youkai. The ability to control the thing we need to survive is one thing, but they can also fly (putting them out of reach from almost all other Youkai who are melee, hint: that would be Vampires, Werewolves, Snowpeople, Neko, mermaids, lamia, and humans) They can also disable their air competitors (the succubus) with their air powers. They can disable a Youkou with their air powers. They can effectively make a tornado of water to disable vampires. I mean there isn't much that isn't overpowered about them. Coupled with their lack of an obvious weakness, they are certainly a worthy adversary.
Granted, any Youkai who can hit something in the air from the ground would certainly count as a weakness for them.
(Screw Fairies) Finally the Lamia. Basically the psychic power of Lycans, but with more staying power and much more refined. Lamia's ability to control minds of Youkai in their Youkai forms are strong, making the only counter to stay in your weak human form, be human, or be immune to the powers. This makes most of the Youkai obsolete in a fight against them. Additionally they only have the glaring weakness of being blind without glasses, (coincidentally the weakness of Velma from scooby-doo, and that weakness really helped the bad guys). I can say this Youkai is overpowered, unless they could only use their powers when they can see the person. Additionally the person has to make eye contact (no matter how brief) in order for them to be useful. Also they should have the ability to influence decisions rather than out-right control them. At least for the sake of this RP. Currently they share a weakness with Lycans, only being relevant in 1v1s though (or possible 2v1 if they can get the two people to attack each other)
The Youkai are very loosely balanced, and imagination is the real power that will make a Youkai seem overpowered. But by pure analysis I would like to stand by the fact that it is not (and will not be) perfectly balanced.