Pardon my ignorance of this topic.

Q1. In which dimensions $d$ is it the case that, for every natural number $n$, there exists a sphere having exactly $n$ lattice points on it $(d{-}1)$-dimensional surface?

Schinzel's Theorem establishes this in $\mathbb{R}^2$, and Kulikowski's Theorem establishes this in $\mathbb{R}^3$. Is it known in higher dimensions?

Q2. Why is it that the smallest integral radius of a circle (in $\mathbb{R}^2$) centered at $(0,0)$ that has $n$ lattice points on its circumference, is always a multiple of $5$? Or rather: Is there a simple, intuitive explanation for the multiple-of-$5$?

OEIS A046122: $1, 5, 25, 125, 65, 3125, 15625, 325, \ldots$. A046122 link

And this likely follows from answers to the previous two questions:

Q3. Does the multiple-of-$5$ phenomenon occur in higher dimensions as well?

I realize these questions may be answered in the literature I have not encountered, so pointers are welcomed—Thanks!