That's a very good question. Just rececntly heard a similar question on radio.

Usually the pr. ИЗ is used when the country (or federal state within a country with clear borders) or town is meant. So strictly speaking the correct sentence is

Я из Украины. Моя семья из Украины. - any country - Я из России, Я из Германии и т.д. And also Они из Баварии must be used (because Бавария is Bundesland which is a separate region within a larger country Germany). Я из Москвы - because it's a city and the borders are quite clear.

But when they talk about 'some location, mountain region, lake region, river, sea, island, peninsula - where you don't really have borders or it's a part of a bigger geographical body the preposition C must be used in most cases.

Examples where C MUST be used: Я с Урала (some place in the area where Urals mountains are), Я с Волги, Я с Кольского полуострова (peninsula is part of bigger continent), Мы с Камчатки, Он с Аляски (peninsula again), but Я из Сибири. As you get more fluent in Russian you don't have problems with using the correct preposition here.

But some native speakers of Russian use C where it really should be ИЗ - these must be some dialects, especially South of Russia, and regions close to Ukraine, you see this might the influence of the Ukranian language where the form is this - Я з України. Я з Бангладешу. Я з Росії.
And also some speakers with poor knowledge of grammar (there are many of them in Russia and Russian speaking countries) might use C instead of ИЗ.

So the correct usage is Моя семья из Украины. But Моя семья с Украины sounds like a dialectical variant with slight deviation from correct usage. The influence of the Ukranian language in this family is more than obvious, so it's quite normal to speak for them like that.

At the same time Украина (when part of the USSR) was understood as area, region, like a geographical location (not independent country like today), so this C-usage might have been quite common during the USSR. So you see in case with Ukraine you really have several factors at play (Ukranian language, history of the USSR)

2) This is in genitive, correct? - answer - correct.

you can also say Моя семья родом из Украины. Мы родом из Украины - meaning we were born in Ukraine.


3) ..... For example could you say 'я с Америки' ? - no, only Я из Америки (usually that would mean USA, for shortness). But you MUST say Я с Гавайских островов - the isles are understood as some geographical body, or Я с Аляски (peninsula as part of a bigger continent)


I hope it all makes sense, it's something off the top of my head,
I can elaborate on that and provide more and better examples if you need.
As I said, linguistically, this is an interesting question.