Does computability according to Church-Turing thesis include side effects?
The Church-Turing thesis says that Turing machines capture precisely the effectively calculable functions from natural numbers to natural numbers. It says little about what happens when we attach a robotic arm to a computer.

Any kind of machine needs to represent numbers in some way, obviously since numbers are abstract entities. A Turing machine represents them with 0's and 1's written on a tape. In $\lambda$-calculus we represent them with Church numerals. If we attached a robotic arm to an iPhone, we could represent them with a suitable number of up-down movements by the arm. The question of representation is separate from the question of side effects. In particular, the question of what can be computed is separate from how the results can be communicated. Lamenting about the fact that $\lambda$-calculus has no input/output, for instance, would be misguided.

There is a well understood theory of computational side effects. According to this theory, Turing machine have certain side effects. They carry state in their cells, that is a side effect. We can implement quoting on Turing machines, that is the side effect which converts a closure to source code. Turing machines with oracles can be seen as having the "input" side effect (known as a reader monad in functional programming).

In summary, the Church-Turing thesis is largely independent of what computational effects might be present in the computational model. This is so because the Church-Turing thesis is about numer-theoretic functions, not about how computers interact with their environment.