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.
It should be 勺子 (or just 勺 as a measure).
调羹 has similar meaning with 勺子, but is partial to the spoon used for eat soup.
And 调羹 usually used in southern dialect.
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