Intransitive verbs describe a change that people/things undergo from my understanding and do not take a direct object. I've also read that the te-form of intransitive verbs with いる describe a state that holds after said change occurs.
E.g., 店が開く (the store opens) ---> 店が開いている (the store opens and remains open)
However, the verb 泣く'to cry' is intransitive, yet 泣いている apparently describes the continuous action of crying (instead of a state)?
IMHO, this really has nothing to do with transitive vs intransitive verbs at all. In my view there is actually a completely separate conceptual distinction between verbs in Japanese, which you need to look at instead. There are:
The ている-form really implies different things depending on which of these two categories of verbs you're dealing with. For the first type of verbs, the action can start, then continue for a while, then stop, so it's possible to be "in the middle of" doing that action at some point in time, so for these verbs, ている indicates you are in the middle of doing that action, and is basically equivalent to the present progressive form in English ("is (verb)ing").
However, state-change verbs in Japanese are basically "instantaneous". That means, they don't have a duration, and you can't really ever be "in the middle" of doing them (you either haven't done it yet, or it's already been completed). Therefore, ている can't mean you're in the middle of it, because that makes no sense, so what it means for these verbs is you are "presently in the state that results from having done (verb)". In English, this often corresponds to the present perfect form ("has (verb)ed").
How do you tell what kind of verb you're working with? Well, in many cases it's fairly obvious. In cases where it isn't, you should consider whether the subject is in a distinctly different state after the verb has finished than they were before it started. If they are, then it's probably a state-change verb. If they aren't, then it's probably a normal activity verb.
So in this case, 開く is clearly a state-change verb (it describes the transition from being "closed" to being "open"). On the other hand, with 泣く, after you've finished crying, are you in a different state than you were before you started crying? Not really. And it's reasonable to think of crying as an activity that can go on for some time, so this is pretty clearly not a state-change verb. It's an activity.
I've also read that the te-form of intransitive verbs with いる describe a state that holds after said change occurs.
As a native speaker, I can say that this definition is correct but too narrow in scope; the case you raised with 泣いている is perfectly valid.
I would map "ている" to english persent-perfect form, "has been" "have been". A quick check to see if you can use ている would be to add もう２時間も "for two hours" and see if it makes logical sense.
店はもう２時間も開いている The store's been open for TWO HOURS! (to someone complainig that the store's closed and can't go)
彼女はもう2時間も泣いている She's been crying for two hours.
In short, your usage is OK. Don't take the textbook too literally.
