1) Expose the environment as virtual folders (like /proc in linux). So that an object in memory coincides with a file in a directory, that in turn would work like a package. For example "ls" would live under /env/fs/ls (and /env/ls would be in "path" for symbol evaluation).
2) Expose the dom, always as files (/env/dom). Make it a plugin. Add a lisp repl to every webpage.
Unfortunately, I haven't worked with Firefox enough to be able to figure out what the problem is. I'll keep looking into it, but if anybody can figure this out, I will be much indebted to them.
If it runs in the browser, it's not an operating system. Nice April 1 joke.
Because it runs in the browser. See, for instance:
or my own
Is the concept ridiculous? Probably. But time transforms all things. Many things that were once ridiculous are no longer so. I'm eager to see how things on this line of thought turn out.
The other part is that the notion of an operating system is not typically tied to a turing machine. An operating system is essentially a set of libraries for any language with appropriate bindings. The language being an abstraction of the underlying machine. To show an equivalence between an OS and browser you list the properties of and operations on a browser and that of an operating system and show there is an isomorphism from one to the other.
Anyways I agree with your conclusion, just not with your argument route. As browsers increase in power they will be indistinguishable from the OS in terms of functionality they expose to the consuming language. The current OSs approach from the other end.
The point being made is not that a browser is an OS, but that something using the facilities of a browser to implement all the operations and environment of an OS is... an OS. This seems almost like the duck typing versus strong typing approach - one view is that it's the behaviors that define the identity of something, the other is that it's the inheritance & implementation.
 Alternatively since it's April Fools, I'll argue that you're interrogating this from the wrong angle and presenting an essentialist view of OS-hood.