Do all words have a part of speech? The closest counterexample I can think of is yes. The dictionary says its supposed to be an adverb but it doesn't really strike me as something that modifies a verb. Are there any words in common use that simply don't have a part of speech? Or, alternatively, are there any parts of speech that they just don't teach in school because they're too abstract to the children learning their parts of speech?
It is intended that all words have a part of speech.
As for your second question - "are there parts of speech they don't just teach in school?" Traditionally, The eight parts of speech are: nouns, verbs, pronouns, adjectives, adverbs, prepositions, conjunctions, and interjections. Students learn all of these.
However, depending on what we're doing, we can use a lot more different types of speech. For example, for my natural language processing project, I used 44 different types of speech (including some like "numbers" and "periods")
Strictly speaking, words don't have parts of speech or indeed any properties at all.
Look at this from the point of view of Luciene Teniere's 'valency grammar', in which a sentence is a sequence of slots with a function, and constructed by dropping appropriate fillers (words or phrases) into the slots. Very briefly, "cats catch mice" has three slots - subject, filled by a doer (cat). Verb, filled by what is done (catch), and object (mice), that is filled by the victim of what is done. If the cats are big hairy salivating ones, then "big hairy salivating cats" would fill the object slot.
The point here is that the 'noun' characteristics of 'cats' do not come from any quality of the word itself, but from the slot it fills in the sentence and how it functions there. This can be seen when 'wrong' words are dropped into slots. A gerund? An -ing verb dropped into an object slot. Elsewhere on this site there is a question about adjectives being used as nouns, with 'The successful are those who strive' as an example. I pointed out that 'the sucessful' is a nominal adjective rather than a noun, but that invites the question "what is a nominal adjective?" It is an adjective being dropped into the object slot of a sentence where we would expect to see a noun.
By putting these words in object slots they have to function as a name of something - become a noun - so we have the same orthographic word functioning as different parts of speech depending which sentence slot it is put in.