I realised that my previous version of this answer was irrelevant to the question (though what was said was accurate, as far as it went).
I think you can make the desired conclusion if you resort to modular representation theory, specifically, the theory of blocks with cyclic defect group (for this question we only need the cyclic defect theory with defect group of order $p$, which was done by R. Brauer himself).
In a block with cyclic defect group, all decomposition numbers are $0$ or $1$. This means that every irreducible character occurs with multiplicity $1$ in some projective indecomposable module (over an appropriate local ring of characteristic zero).
Let $\chi$ be an irreducible character of a finite group $G$ which has order divisible by $p$ and not $p^{2}.$ Let $\theta$ be the character of a suitable projective indecomposable module as above, with
$\langle \theta, \chi \rangle = 1.$
It is a theorem of W. F. Reynolds (a consequence of Brauer's characterization of characters), that $\theta$ is a $\mathbb{Z}$-linear combination of characters, each of which is induced from
a Brauer elementary $p^{\prime}$-subgroups of $G$.
This means that $\chi$ occurs with multiplicity $1$ in the difference
$\alpha - \beta$, where $\alpha$ and $\beta$ are characters of representations which are visibly respectively realizable over $\mathbb{K} = \mathbb{Q}[\omega]$, where $\omega$ is a primitive
$m$-th root of unity and $m$ is the $p^{\prime}$-part of the exponent of $G$ ( in this case, just $\frac{e}{p},$ where $e$ is the exponent of $G$).
The Schur index $m_{\mathbb{K}}(\chi)$ divides both $\langle \alpha, \chi \rangle$ and $\langle \beta, \chi \rangle$ by the defining properties of the Schur index . Since $\langle \alpha - \beta, \chi \rangle = 1,$ we conclude that $m_{\mathbb{K}}(\chi) = 1.$
In the case of $G = {\rm GL}(2,p),$ this gives a positive answer to part $1$ of the question (every element of order divisible by $p$ has central $p^{\prime}$-part, and every element of order $p$ is conjugate to all its powers. Hence all character values on $p$-singular elements take values in the field $\mathbb{K}$ as above. and clearly all character values at $p$-regular elements take values in $\mathbb{K}$
(recall that $p$-regular means of order prime to $p$ and $p$-singular means of order divisible by $p$)).
Well, I suppose that strictly speaking, the answer as it stands does not now cover the irreducible characters of degree (divisible by) $p$,it only deals with the irreducible charcters in $p$-blocks of defect $1$. But for any irreducible character $\chi$ in a $p$-block of defect zero of any finite group $G$, the irreducible character $\chi$ is afforded by a projective $RG$-module, where $R$ is a suitable local ring of characteristic zero with residue field of characteristic $p$. Using the result of Reynolds, and arguing as above, we obtain $m_{\mathbb{K}}(\chi) = 1$, where $\mathbb{K}$ is the cyclotomic field extension of $\mathbb{Q}$ generated by a primitive $m$-th root of unity, with $m$ being the $p^{\prime}$-part of the exponent of $G$. In this case, $\chi$ vanishes on elements of order divisible by $p$, so that all character values taken by $\chi$ are in $\mathbb{K}.$
