A few months ago I posted a question about analytic functions that I received from a bright high school student, which turned out to be studied and resolved by de Bruijn. Based on this positive resolution, I thought I might try my luck again and list three further questions that this student asked which do not seem to be trivially resolvable.

1. Does there exist a smooth function ${f: {\bf R} \rightarrow {\bf R}}$ which is nowhere analytic, but is such that the Taylor series ${\sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n}$ converges for every ${x, x_0 \in {\bf R}}$? (Of course, this series would not then converge to ${f}$, but instead to some analytic function ${f_{x_0}(x)}$ for each ${x_0}$.) I have a vague feeling that perhaps the Baire category theorem should be able to resolve this question, but it seems to require a bit of effort. (Update: answered by Alexander Shaposhnikov in comments.)
2. Is there a function ${f: {\bf R} \rightarrow {\bf R}}$ which meets every polynomial ${P: {\bf R} \rightarrow {\bf R}}$ to infinite order in the following sense: for every polynomial ${P}$, there exists ${x_0}$ such that ${f^{(n)}(x_0) = P^{(n)}(x_0)}$ for all ${n=0,1,2,\dots}$? Such a function would be rather pathological, perhaps resembling a space-filling curve. (Update: solved for smooth ${f}$ by Aleksei Kulikov in comments. The situation currently remains unclear in the general case.)
3. Is there a power series ${\sum_{n=0}^\infty a_n x^n}$ that diverges everywhere (except at ${x=0}$), but which becomes pointwise convergent after dividing each of the monomials ${a_n x^n}$ into pieces ${a_n x^n = \sum_{j=1}^\infty a_{n,j} x^n}$ for some ${a_{n,j}}$ summing absolutely to ${a_n}$, and then rearranging, i.e., there is some rearrangement ${\sum_{m=1}^\infty a_{n_m, j_m} x^{n_m}}$ of ${\sum_{n=0}^\infty \sum_{j=1}^\infty a_{n,j} x^n}$ that is pointwise convergent for every ${x}$? (Update: solved by Jacob Manaker in comments.)

Feel free to post answers or other thoughts on these questions in the comments.