"Do you know any good apps?"
This sentence sounds very unnatural. I think the problem is from translating know literally to 知道, but actually it works only in some cases. e.g. I know him is not 我知道他 but 我认识他.
To ask for advice like this, it could have been：
Because the answer is not accepted without 有, the English question is being taken to focus on knowing whether such apps EXIST, rather than one being familiar with (any) apps. Sort of: Are there any good apps that you know of? Perhaps the use of 知道 without 有 might mean something like: Do you know the ins and outs of any good apps? In other words, is might imply a deeper level of knowledge. Can anyone shed any light on this?
Because there is 什么, which should already be sufficient for asking which apps there are.
Because 什么 already indicates that is a question. Same thing with 谁 for who or 几 or 多少 etc for how much.
The word for "app" I kept seeing and hearing in Taiwan and the mainland was "APP" pronounced "ei pi pi" but I didn't know 应用 while I was over there, so maybe it's even more common?
I'm with you. I know the word 应用 but have never, ever heard it used in this context. Like you, I've only heard native speakers say "a-p-p" for app. Also, "app" is not one of the definitions of 应用 in my Chinese dictionary.