Will the pool water give our cat any health problems?
Most likely not. If the water is clean and you're not using an excessive amount of chemicals (like chlorine), there shouldn't be any problem. If in doubt, you can confirm with your vet, potentially even bringing a water probe, but I think this would be exaggerating the whole thing. As long as your cat is ...
Bettas are anabantoids, commonly known as labyrinth fish after the labyrinth organ, which is a kind of lung-like extension of the gill tissue which can absorb oxygen from air.
For labyrinth fish, occasional surface gulping is quite normal and is a supplementary breathing action. This doesn't mean the fish isn't suffering from ammonia, nitrite or chlorine ...
You might want to try a flotation jacket for the dog, they are not too expensive and might give her a bit more confidence in the water (should certainly support her in the water) and allow her to learn how to use her legs
While I cannot ask as a comment, and you mentioned trying everything, have you tried Erythromycin yet?
In the photo, I noticed the fins seemed ragged and translucent in places, typical of either fin damage (sharp objects/other fish biting) or Fin Rot, a bacterial infection.
Erythromycin treats bacterial infections, including Fin Rot.
Other remedies like ...
You wind up unintentionally taking in some of that water every time you swim. It isn't exactly up to potable-water standards, but it's better than the puddles your cat is also drinking from when you aren't watching. Don't worry about it from that point of view.
Do try to make sure that if the cat actually falls into the pool there's a place where she can ...
As far as I know dogs cannot physically close their nostrils, among mammals only the aquatic ones possess such mechanisms.
But completely sealed nostrils are not actually that necessary because water entering just the nose itself is not that dangerous - it's the deeper parts of the respiratory system that must be protected - and such protection mechanisms ...
You need to take your kittens to the vet. Dehydration is very dangerous to kittens and it might be fatal very quickly, if not treated in time.
The best thing is to avoid chlorine in a swimming pool where animals are likely to drink water. There are better and less harmful chemicals you can use, like hydrogen peroxide.
And there is the obvious solution of ...
Based on your response above, I'd definitely recommend a working breed and make them big so they can handle the distance elements.
Some examples might be:
German or Belgian Shepherds
A medium drive german shepherd would be perfect, enough drive to want to work/...