Зовут ее Светланою

Yes, it's её not ее there, but Russians tend not to write or type ё letter and substitute it with just е, that's because we're lazy. So if you see a word with ё is spelled with е instead, don't be confused.

Светланою is a poetic form of instrumental case of Светлана, which is Светланой. Poets like to replace the й at the end of a word with ю if the й is a part of a case ending which -ой is.

Как в песенке одной

Yes, песенке is in prepositional (locative) case and yes, the prepositional of песня is песне. The nominative of песенке is песенка though. So what is the difference between песня and песенка you might ask? Well, the difference is really subtle, basically it's the same thing, but we use -енк and some other suffixes to form so called уменьшительно-ласкательная форма of words. It's called diminutive in English.

Sometimes those dimunitives really convey the sense that is described in the wiki-article I provided, sometimes not though. For example, I think the poem author chose to use the dimunitive of песня, which is песенка, just to achieve a smooth rhytm.

I hope this helps