Learn Russian: Это чья-то сумка.

чья-то (someone's) acts like an adjective, so it has to match the gender of the noun it belongs to. Since сумка (bag) is feminine, you use the feminine form чья-то instead of the masculine чей-то.

If we put this in a different case, does the -то part change?

No, the -то suffix acts like a tail that never changes. Only the first part of the word changes for case and gender. For example, if you wanted to say "in someone's bag" (prepositional case), it would be в чьей-то сумке.

сумка is feminine, so why does the sentence start with the neuter word Это instead of Эта?

When you use Это to mean "This is..." to introduce or point out a thing, the word is fixed and doesn't change, regardless of the noun's gender. You would only use the feminine Эта if it was attached directly to the noun, like in the sentence Эта сумка чья-то (This bag is someone's).

Could I use the genitive of кто-то (someone) and say сумка кого-то?

While people would understand you, чья-то сумка is the natural way to say it. Because Russian has a specific possessive word for "someone's" (чей-то and its forms), native speakers use it instead of saying "the bag of someone".