Learn Polish: Kobieta rozumiała problem, a oni nic nie rozumieli. - The woman understood the problem, but they understood nothing.

This is the e/a alternation you are focusing on right now! For verbs ending in -eć like rozumieć (to understand), the e in the stem changes to a in most past tense forms. So, ona rozumiała (she understood) and on rozumiał (he understood) both use the a. But the e stays in the masculine personal plural form: oni rozumieli (they understood).

Is problem in the accusative case here?

Yes, it is the direct object of the verb rozumiała, so it takes the accusative case. Because problem is a masculine inanimate noun, its accusative form looks exactly the same as its dictionary form.

Could I use ale instead of a for the word but in this sentence?

You could, but a is much better here. Polish often uses a to contrast two different subjects or situations. Here, we are contrasting the woman (who understood) with the group (who did not). The word ale is typically used when the second part of a sentence introduces an unexpected contradiction.

Why does the sentence say nic nie rozumieli? Is that a double negative?

Yes, Polish strictly requires double negation. When you use a negative pronoun like nic (nothing), you must also put the negative particle nie directly before the verb.