When a recipe asks for 1 cup of sugar, should I assume powdered or tiny crystals form?
"Sugar" without any other description refers to granulated sugar. Granulated means crystals somewhat larger than the crystals in caster sugar, which is another commonly used crystal sugar in the UK. In other countries these smaller crystals may also be called superfine sugar or baking sugar. You can generally use granulated or caster/baking/superfine sugar interchangeably, but the volumetric measurement (spoons or cups) will be slightly different since granulated sugar is coarser than caster sugar. You might find this Sugar Conversion Tool helpful.

If the recipe wants you to use the powdered form of sugar, it will specify powdered, confectioners or icing sugar, which are all the same thing.