Do they sound natural in: "Was your question addressed/aimed/directed at me?" and "His spiteful remark was aimed/addressed/directed at Jane's husband."? Does "address" sound unnatural in these sentences?
No, it sounds okay with the correct preposition. It's more a question of style: "aimed at" implies a metaphorical violence, so it matches better with "spiteful remark." "Addressed to" and "directed at" are neutral in tone.