If I'm referring to "product number 3" but want to shorten it to just "number 3" - would it be proper to use 数 or 番号?
Product Number 3 - 商品3番, number 3 - 3番
数(かず) means a number, it cannot be used as a counter.
番号 means a number that is assigned to the counter. e.g. この商品の番号は3です。But it cannot be used as a counter either.