If it's truly closed--not on your tintype. To run entropy backwards, you need to have energy made available from outside.
I have never been able to understand Shannon's notion of information entropy, and not for lack of trying. If you understand it, more power to you. But I think the old-fashioned notion of entropy is the only one relevant to biological systems. DNA is not bits in an etherial bit plane, and you'd be well advised, in my opinion to stear clear of the notion of informational entropy. DNA creates proteins, and whether having more or less more-or-less complicated proteins is an increase in informational entropy does not tell you much--but the tendency to meld issues in energy entropy and informational entropy can lead you into forbidding things that quite obviously could and have occured for not very sound theoretical reasons.
Information theory provides the ability to measure the amount of entropy in any system. In thermodynamics, it is frequently much easier to derive the entropy values indirectly. Incidentally, DNA is very much related to information theory and is a fine example of mathematical coding in biological systems. Information theory is extraordinarily powerful, but for the practice of engineering one rarely needs to go to that level to get the answer one is looking for.
The greatest danger of NOT tying information theory and thermodynamic entropy together is that people take them to mean different things and ascribe characteristics to thermodynamic entropy that would be clearly invalid if you understood information theory. There is a lot of evidence that this is in fact going on when I see words such as "complexity" to define entropy in thermodynamics, when that is a nonsense definition under the formal information theory definition of entropy. They do not use different definitions; it is just easier to fudge correct results with invalid definitions in thermodynamics than it is in information theory (assuming you aren't doing real work with it, that is).
You are correct about one thing though: "entropy" as it is actually defined is not an easy to grasp or intuitive concept. In this case though, the "made-for-TV" definitions are proving to actually be detrimental to people's understanding of it when they try to apply it.
I'm sticking to the old fashioned notion in my discussions. Also, not for lack of trying, I could not do otherwise. I agree with you.