Ida is rather pretty, though not a personal fave. Sounds old, but not in a bad way.
I think
Ida would be cute for a little girl, but would also mature well.
I've never heard it pron. EE-da, always EYE-da, but although either is fine, I think EE-da is prettier.
Doubt it would cause any pron. problems, you'd probably just have to correct people once. I don't know, maybe in other places EE-da is the common pron. And I'd say leave the sp. as is.
Really have no idea what kind of mn's you would like, but a few combos are:
Ida CarolineIda JuneIda LorraineIda EvelynIda EleanorIda ClareIda KatherineIda CharlotteIda HelenIda JosephineIda JulietIda LavenderIda LouiseIda MiriamIda Victoria