Other examples may clarify why it does not work: I do running? I do biking? It DOES communicate the same thing. But that does not mean it is correct english. I bike. I run. I read. I will do some biking. I can do some running. I should do some reading. All these are more correct English. (not necessarily better at at communicating.)

Yes, you may correctly say "do some reading". This form is different from "do reading", which is not correct.

You can use 'do reading'. for example: "I need to do some reading on the subject" or "The homework for tonight is to do some reading'