Learn Italian: Non sanno la risposta a questa domanda. - They don't know the answer to this question.

Yes, sanno is the "they" form of the irregular verb sapere (to know). A regular verb would usually end in -ono, but sapere is irregular, so you just have to memorize sanno for "they know". The subject loro (they) is left out here because the ending makes it clear who is doing the action.

Could I use the verb conoscere here instead of sapere?

No. In Italian, there are two verbs for "to know". Sapere is used for knowing facts, information, or how to do something. Since the answer to a question is a piece of information, we use sapere. The other verb, conoscere, is used for being familiar with people, places, or complex topics.

Why do we use the word a after risposta?

Just like in English where we say "the answer TO a question", Italian uses the preposition a (to) to connect la risposta (the answer) with the specific question it belongs to.

Why is it questa domanda and not questo domanda?

Questa means "this" and it must match the gender and number of the noun it describes. Because domanda (question) is a feminine singular word, you must use the feminine singular form questa instead of the masculine questo.