Which word should I use for "spoon", as a noun or as a measure.
form the function point of view, 勺子, 瓢 and 调羹 are the same, they are a sort of object used to get water.
from the use cases point of view, 瓢 is used to get water only and not suitable to be appear on a table since it's too large, and 勺子 is used to get soup, 调羹 is used to get whatever food in your bowl. so typically in size 瓢 > 勺子 > 调羹, BUT, there's no limitation that you must use 勺子 to get soup, it's just the table manner.
BTW, 瓢 originally stands for an tool made of very old calabash which is used to get water. when a calabash gets old, it will be very hard, so people cut it in the middle and a calabash becomes two 瓢.
勺儿 sháo er
勺子 sháo zi4
You can both use it as the name of a tableware or other instruments resembling the shape of a spoon.
You can also use it as the a measure.
You can also use these two in the scenarios you mentioned. However, they are more formal, and normally used in written or in a setting formal oral language are required (e.g. TV program).
Mandarine spoken in southern and mid China:
勺子 sháo zi3