For some conditional limit theorems see cambridge.org/core/journals/journal-of-applied-probability/…

In your example $\delta_{a_n}\longrightarrow \delta_0$ weakly (by the continuity theorem, since $\Phi_n\longrightarrow \Phi_0$ pointwise). Your reminder is incorrect: $\mu_n\longrightarrow \mu$ weakly iff $\mu_n(E)\longrightarrow \mu(E)$ for all $\mu$-continuity sets $E$. And here $E$ is clearly not a continuity set of $\delta_0$ since $0\in \partial E$.

It is quite conceivable that similar structures as above appear also for more complicate $T$ -functions (but that is only a guess). If the final sums can be expressed via cycle counts generating functions might simplify a lot.

Your interpretation of the "coefficient operator" is correct. The first formula follows in a (more or less) routine way from the known generating function of the cycle type of (uniform) self mappings of $[n]$, the formula for $x$ then only needs basic properties of coefficient extraction.

(I think) your sum can be simplified using generating functions. It can be shown that $N_{d,k}=d! [y^dt^k] e^{(d-1)y+(t-1)\frac{y^2}{2}}$ and that consequently your $x$ can be expressed as $$x=r^{|E|} d! [y^d] e^{(d-1)\frac{s_1}{r}y+\big(\frac{s_2}{r}-\frac{s_1^2}{r^2}\big)\frac{y^2}{2}}$$ (essentially a Hermite polynomial evaluated at an imaginary argument). Is that of interest or are you content with what you have?

Your final inequality appears to be violated for $n=3, a_1=a_2=a_3=\frac{1}{2}$ ?

@Fedor Petrov: No, it can be derived independently, using a differential equation/undetermined coefficients. Thanks, I have added that information.

An elementary proof of the Stone-Weierstrass theorem is given in ams.org/journals/proc/1981-081-01/S0002-9939-1981-0589143-8/‌​…

Indeed, most of these results had already been obtained (in a slightly different guise) by J.F.C. Kingman in 1982, see inference.org.uk/sustainable/… . A generalization to more general distributions was given in arxiv.org/pdf/0809.4233.pdf

See this paper of Fill: citeseerx.ist.psu.edu/viewdoc/…

Another observation: the ($\mathbb{Z}/2\mathbb{Z}$-) dimension of the kernel of $M_n \bmod 2$ seems to be $\frac{2^n+2}{3}$ for even $n$ and $\frac{2^n+1}{3}$ for odd $n$.

These things are well known. Look up "singular inversion" and "singular expansion", see e.g. Lemma IV.3 and Thm VI.6 in the book "Analytic Combinatorics" by Flajolet and Sedgewick.