is ζάχαη here in genitive? or nominative? "a unit of something", is it always unit first and nominative of that something after? A glass of water=ένα ποτήρι νερό, correct?
It's in accusative actually. There is a long set of rules about noun modifiers which i guess it's not easy to include in the course, but what you need to know about this sentence is that when the first noun is a unit of measure for the second noun, then the second noun should be in the same case as the first noun. In this sentence there are two clauses actually, the subject and the verb of the first clause is omitted. In its full form it would be "Εγώ θέλω ένα κουτάλι ζάχαρη, παρακαλώ" (I want a spoon of sugar, please), so "κουτάλι ζάχαρη" is the object, hence accusative case, but in this set of nouns it happens to be exactly the same in nominative and accusative.
Please notice that the same set of nouns can be either a same case modifier or a different case modifier (the usual case used on the second noun on a different case modifier is the genitive), depending on the meaning we want to convey according to the rules I mentioned above. For example "ένα ποτήρι κρασί" (a glass of wine) is a same case noun modifier (ομοιόπτωτος ονοματικός προσδιορισμός) and the first noun is a unit of measure of the second noun (a glass full of wine), whereas "ένα ποτήρι κρασιού" (a wine glass) is a different case noun modifier (ετερόπτωτος ονοματικός προσδιορισμός) and the second noun describes the use of the first noun (a glass for wine). Similarly, "ένα φλιτζάνι τσάι" = a cup of tea, vs "ένα φλιτζάνι τσαγιού" = a teacup, etc.
Small spoons are "κουταλάκι του γλυκού", soup spoons are "κουτάλι της σούπας".