Actually according to the naruto wiki, the Toad Sages have the ability known as Frog Kata and the ability of the snake sage mode users is the ability breathe life into inorganic substances known as Sage Art: Inorganic Reincarnation, but its not the power of the White snake.

at least thats what the wiki says.

but yeah those all sound reasonable. honestly I'm more confused as to how water-based summons even work: they don't really work on land, so you have to say near water to make them work, or produce enough for them to swim through, or they have to do it themselves as soon as they're summoned. perhaps it best to make summoning contract archetypes rather than individual ones.

as for the seal....maybe? I'm not sure if any paper tag is innocuous considering one of the most common tags made are basically grenades. but sure whatever.