Нет, это звучит очень странно. Так не говорят.
You can say: "Ты знаешь как читать?" But 'как' accents on the knowledge about the method
Wouldn't that mean something more like "do you know reading" or "do you know to read"?
Does this mean exclusively in a general sense, or can it be like "can you read" in a particular instance?
It means reading in general sense. "Уметь" is "can" in a sense of "have a skill". And if you know how to read in general, this skill is allways with you even if you can't read something at the moment because it's too dark or something like that.
When we are talking about being able to do something we use "мочь".
I'm not native, but I've got the impression that Americans tend to form this kind of sentence like this: "do you read?".
Would this sentence be acceptable here?
Yes, that doesn't work in English. Seems like the construction another language would do. What's your native language?