The general rule for "only" is to snuggle it up as closely as possible to the word or phrase you intend it to modify. "He only speaks English" can be interpreted as "He does nothing other than speak English," whereas "He speaks only English" should reliably be interpreted as one of "He does not speak anything other than English" or "He is not speaking anything other than English."
If you're interested in conveying something about what languages the subject can speak, however, then you must rely on more than the placement of the word "only", because "to speak" a language can mean either to use that language in speaking or to be able to use it (M-W), as I touched on above. Certainly if someone is speaking a language then it follows that they can do so, but when you add an "only" then it matters which sense of "to speak" applies, and that can be unclear.
The distinction can come from multiple sources, including context, verb form ("is speaking" unambiguously describes current usage, not ability), and idiom (omitting the "in" is more idiomatic when you are talking about a person's ability to speak a language). Sometimes, however, it's simply ambiguous.
With respect to your specific propositions:
I would simply say "He is speaking English," but you could emphasize that the subject is speaking English exclusively by saying "He is speaking only English." This does not convey anything about the subject's ability to speak other languages; for that I would probably choose an altogether different formulation, maybe like so: "He is restricting himself to English."
Similar applies here: you are trying to convey two different, but related, ideas at the same time. "He speaks only English" could convey either one of them, but it can be interpreted as describing only the subject's current language. If you want to reliably convey both ideas in the same sentence then you could say something like "He is speaking his only language, English."
The sentence you proposed could be interpreted as you describe, as could "Only he speaks only English," and both probably would be interpreted that way. Nevertheless, there is again an ambiguity concerning whether the sentence describes the subject's capability or merely his current action. Since in this case you are describing capability, one of the clearer ways I can think of to state the proposition would be "Only he cannot speak any language other than English."