Learn Polish: Wolisz malowanie czy rysowanie? - Do you prefer painting or drawing?

These are verbal nouns, which describe the action of the verb. For verbs ending in -ować, like malować (to paint) and rysować (to draw), you simply drop the final -ć and add -nie. They behave exactly like regular neuter nouns.

The verb woleć requires an object in the accusative case. Why haven't the endings of malowanie and rysowanie changed?

Because verbal nouns ending in -nie are neuter nouns. For all neuter nouns in Polish, the accusative case form is exactly the same as the nominative (dictionary) form. They are technically in the accusative case here, but the spelling doesn't change.

Could I use these words to say "I bought two paintings and a drawing"?

Not usually. Verbal nouns refer to the action of doing something (the act of painting or drawing). If you mean the physical objects, you would use different nouns: obraz (a painting) and rysunek (a drawing).

Why is czy used here for "or" instead of albo?

Czy is used for "or" when you are asking a question that offers a direct choice between options. Albo is used for "or" when you are making a statement.