Are all turing complete languages interchangeable
Turing-completeness says one thing and one thing only: a model of computation is Turing-complete, if any computation that can be modeled by a Turing Machine can also be modeled by that model.

So, what are the computations a Turing Machine can model? Well, first and foremost, Alan Turing and all of his colleagues were only ever interested in functions on natural numbers. So, the Turing Machine (and the λ-calculus, the SK combinator calculus, μ-recursive functions, …) only talk about the computability of functions on natural numbers. If you are not talking about a function on natural numbers, then the concept of Turing-completeness doesn't even make sense, it is simply not applicable.

Note, however, that we can encode lots of interesting things as natural numbers. We can encode strings as natural numbers, we can encode graphs as natural numbers, we can encode booleans as natural numbers. We can encode Turing Machines as natural numbers, which allows us to create Turing Machines that talk about Turing Machines!

And, of course, not all functions on natural numbers are computable. A Turing Machine can only compute some functions on natural numbers, the λ-calculus can only compute some functions on natural numbers, the SK combinator calculus can only compute some functions on natural numbers, …. Surprisingly (or not), it turns out that every model of computation (that is actually realizable in our physical universe) can compute the same functions on natural numbers (at least for all the models we have found up till now). [Note: obviously, there are weaker models of computation, but we have not yet found one that is stronger, except some that are obviously incompatible with our physical universe, such as models using real numbers or time travel.]

This fact, that after a long time of searching for lots of different models, we find, every single time, that they can compute exactly the same functions, is the basis for the Church-Turing-Thesis, which says (roughly) that all models of computation are equally powerful, and that all of them capture the "ideal" notion of what it means to be "computable". (There is also a second, more philosophical aspect of the CTT, namely that a human following an algorithm can also compute exactly the same functions a TM can compute and no more.)

However, none of this says anything about 


how efficient the various models are
how convenient they are to use
what else they can do besides compute functions on the natural numbers


And that is precisely where the differences between different models of computation (and programming languages) come into play.

As an example of different performance, both a Random Access Machine and a Turing Machine can copy an array. But, a RAM needs $O(size_{array})$ operations to do that, while a TM needs $O(size_{array}^2)$ operations, since it needs to skip across $size_{array}$ elements of the array for copying each element, and there are $size_{array}$ elements to copy.

As an example for different convenience, you can just compare code written in a very high-level language, code written in assembly, and the description of a TM for solving the same problem.

And your light switch is an example of the third kind of difference, things that some models can do that are not functions on natural numbers and thus have nothing to do with Turing-completeness.

To answer your specific questions:


  But can any program written in a Turing complete language be re-written in another?


No. Only if the program computes a Turing-computable function on natural numbers. And even then, it might need a complex encoding. For example, λ-calculus doesn't even have natural numbers, they need to be encoded using functions (because functions is the only thing λ-calculus has).

This encoding of the input and output can be very complex, as can expressing the algorithm. So, while it is true that any program can be rewritten, the rewritten program may be much more complex, much larger, use much more memory, and be much slower.


  What if my Assembly has a LIGHTBUTTON opcode? I physically can't emulate that language on a system (language) without a lightbulb.


A lightbulb is not a Turing-computable function on natural numbers. Really, a lightbulb is neither a function nor a computation. Switching a lightbulb on and off is an I/O side-effect. Turing Machines don't model I/O side-effects, and Turing-completess is not relevant to them.


  On arbitrary real numbers.


Turing-completeness only deals with computable functions on natural numbers, it doesn't concern itself with real numbers.

Turing-completeness is simply not very interesting when it comes to questions like yours for two reasons:


It is not a very high hurdle. All you need is IF, GOTO, WHILE, and a single integer variable (assuming the variable can hold arbitrarily large integers). Or, recursion. Lots and lots and lots of stuff is Turing-complete. The card game Magic: The Gathering is Turing-complete. CSS3 is Turing-complete. The sendmail configuration file is Turing-complete. The Intel x86 MMU is Turing-complete. The Intel x86 MOV instruction is Turing-complete. PowerPoint animations are Turing-complete. Excel (without scripting, only using formulas) is Turing-complete. The BGP routing protocol is Turing-complete. sed is Turing-complete. Apache mod_rewrite rules are Turing-complete. Google for "(accidentally OR surprisingly) turing complete" to find some other interesting examples. If almost everything is Turing-complete, being Turing-complete stops being an interesting property.
It is not actually necessary to be useful. Lots of useful stuff isn't Turing-complete. CSS before version 3 isn't Turing-complete (and the fact that CSS3 is isn't actually used by anyone). SQL before 1999 was not Turing-complete, yet, it was tremendously useful even then. The C programming language without additional libraries doesn't seem to be Turing-complete. Dependently-typed languages are, more or less by definition, not Turing-complete, yet, you can write operating systems, web servers, and games in them.


Edwin Brady, the author of Idris, uses the term "Tetris-complete" to talk about some of these aspects. Being Tetris-complete isn't rigorously defined (other than the obvious "can be used to implement Tetris"), but it encompasses stuff like being high-level enough and expressive enough that you can write a game without going insane, being able to interact with the outside world (input and output), being able to express side-effects, being able to write an event loop, being able to express reactive, asynchronous, and concurrent programming, being able to interact with the operating system, being able to interact with foreign libraries (in other words: being able to call and be called by C code) and so on. Those are much more interesting features of a general purpose programming language than Turing-completeness is.



You may find my answer to the question you linked interesting, which touches on some of the same points even though it answers a different question.