You may well consider the dependence on $s$ of the solutions to $\sum a_kx^k+s=0$. Assuming you have an isolated positive root, its location will be a differentiable function of $s$ in a small neighborhood. You will be able to see the sign of its derivative from the usual chain rule. Then it's just a matter of precomposing with $f(t)$.

The OP assigns a rather nonstandard meaning to "chopping off a vertex".

Take a cone over a complicated polytope in $\mathbb R^{n-1}$, i.e. a convex hull of some points of the form $(w_i,0)$ and $({\bf 0},1)$. Then wiggle the points $(w_i,0)$ to $(w_i,\epsilon_i)$. When you remove the vertex $({\bf 0},1)$, the new number of facets is equal to the number of simplices in a regular triangulation of $conv(\{w_i\})$. I think that this could be significantly larger than the number of facets of $conv(\{w_i\})$.

It is very easy to imagine a situation where removal of a vertex results in a significantly more complicated polytope. By the way, I would not call this "chopping off a vertex". That term I would reserve to adding additional inequality that would remove a little piece of the polytope next to a vertex (and give one more face).

You seem to be missing the inequalities $m,n\geq 0$. This will lead to weird inequalities for $m+n,m-n$, and I don't see how you would get products.

It is possible that your sum would lead to something along the lines of mock theta functions, but it is out of my expertise area.

I am not sure if this is exactly what you need though.

arXiv:math/9904126, equation (11) page 14.

I think that if you instead consider the sum for $n\in \mathbb Z$, then you get an easy to write elliptic function. But I am not sure what to do with $n\geq 0$.

I tried the $7$ instead of $5$. For initial values in $[1,100]^2$ there is periodicity.

Certainly, if the result is already known, then it is not worth sweating out the details. I do wonder what happens if one replaces $w$ by nearby numbers. For example, what happens if you do $2\cos(2\pi/7)$ as opposed to $2\cos(2\pi/5)$? What happens if you take a rational number nearby? Can you assure that there is a transcendental number near $w$ such that there is a recursive sequence like this that goes to infinity? What happens for higher order "almost linear" recursions with characteristic roots of length one? This must have been studied, but so far nobody gave a reference.

Can you be more explicit about how you prove your assertions. I am guessing that this is right, but I don't quite follow the argument.

Yeah, it feels like one would need to find some "walls" that can not be crossed. I wonder if one can think of some congruences in the ring $\mathbb Z[x]/\langle x^2+x-1\rangle$ to allow for a better control after five iterations.

We are not passing judgement on you, just on your question. If you have actual research questions in the future, this may well be the site for them.

This does not look research level to me.

An obvious remark is that one should show that the sequence is bounded. This would imply periodicity without any specific information about the period.

Molien series (dimensions of graded pieces) is of course easily computed, but the structure of the ring can be harder to get.

Of course. There is still a concept of Koszul dual, though.