Skill ups come from feeding a creature another with the exact same ability
You'll lose any skill progress if the skill changes when the creature evolves (eg, a lot of the dragons go through progressively stronger nukes, so if you fed a lil' dragon another one and ranked up its baby nuke, it'd lose that progress when it evolved)
In Siren's case, you can start stuffing her full of blue goblins right away, she keeps her same orb change ability from Mermaid-> Siren -> StE (Siren the Enchanter)
For completions sake, there are two other healer girls, for red/black, though they're not 'healers' in the same sense as the others. They're classified as Healer, and have healer style stats (high recovery), but their abilities aren't the same.
Naga->Echidna has a turn delaying ability that's really useful (her leader passive is a nuke every time you match, the inverse of heal on match :V)
Succubus -> Lilith has a poison ability, vital for tackling certain high defense enemies (her leader passive lets you drag orbs around a bit longer)