I've compiled my answers into a pdf. Highlights:

The space $\Theta_n$ of $n\times n$ simply diagonalizable real matrices has components

$$ \Theta_n=\bigsqcup_{r+2s=n} \Theta_{r,s} $$

where $\Theta_{r,s}$ is the simply diagonalizable real matrices with $r$ invariant lines and $s$ invariant planes. The component $\Theta_{n,0}$ has those matrices which are diagonalizable over $\Bbb R$ specifically.

It is a fiber bundle $G/T\to\Theta_{r,s}\to {\rm UConf}(s,{\cal H})$ where $G={\rm GL}(n,\Bbb R)$, $T$ is subgroup of block diagonal matrices with $r$ scalars followed by $s$ scalar multiples of $2\times 2$ rotation blocks, $\mathrm{UConf}(k,M)$ (unordered configuration space) is the space of sets of $k$ distinct points of a manifold $M$, and ${\cal H}$ is the open upper half plane in $\Bbb C$. Using things like orbit-stabilizer, the Iwasawa decompositon $G=KAN$, and the long homotopy exact sequence, we can deduce:

Theorem. $\pi_1(\Theta_{r,s})$ is an extension of $\pi_1(G/T)$ (below) by either the braid group $B_s$ (if $r>0$) or its "alternating braid subgroup" (if $r=0$).

$$ \pi_1(G/T) \,\cong\, \begin{cases} \, \Bbb Z_2^{r-1} & s>0, r>0 \\ \, 0 & s>0, r=0 \\ \, \Bbb Z & s=0, r=2 \\ \, H_r & s=0, r>2 \end{cases} $$

Here $H_r$ is the subgroup of the spin group ${\rm Spin}(r)$ generated by $e_ie_j$ for $1\le i,j\le r$ (wrt a given orthonormal basis), subject to relations $e_i^2=1$ and $e_ie_j=-e_je_i$ ($i\ne j$). For example $H_3=Q_8$ is the quaternion group.

Theorem. The space $\Xi_n$ of all $n\times n$ simply diagonalizable complex matrices has $\pi_1(\Xi_n)=B_n$.

The space of $n\times n$ matrix solutions to $A^m=I$ is a representation variety whose components are orbits under conjugation. There is one orbit for each $n$-dim rep of $\Bbb Z_m$. Any rep can be encoded as a multiset of irreps (up to iso). The irreps of $\Bbb Z_m$ over both $\Bbb R$ and $\Bbb C$ are well-known. It is possible to translate "dimension $n$" and "has order exactly $m$" each into a condition on the irrep multiplicities.

Over $\Bbb C$, each orbit is simply connected. Over $\Bbb R$, the fundamental group of the orbit is either: $\Bbb Z_2^{k-1}$ if not all irreps that appear are 1D (where $k$ is the total number of inequivalent irreps which appear), $H_n$ if there are only 1D irreps and the dimension is $n>2$, or $\Bbb Z$ if $n=2$ with two 1D irreps. (I think I messed this up in the pdf because I'm running out of steam at the moment. I'll fix it later.)

I've compiled my answers into a pdf (updated version). Highlights:

The space $\Theta_n$ of $n\times n$ simply diagonalizable real matrices has components

$$ \Theta_n=\bigsqcup_{r+2s=n} \Theta_{r,s} $$

where $\Theta_{r,s}$ is the simply diagonalizable real matrices with $r$ invariant lines and $s$ invariant planes. The component $\Theta_{n,0}$ has those matrices which are diagonalizable over $\Bbb R$ specifically.

It is a fiber bundle $G/T\to\Theta_{r,s}\to {\rm UConf}(s,{\cal H})$ where $G={\rm GL}(n,\Bbb R)$, $T$ is subgroup of block diagonal matrices with $r$ scalars followed by $s$ scalar multiples of $2\times 2$ rotation blocks, $\mathrm{UConf}(k,{\cal H})$ (unordered configuration space) is the space of sets of $k$ distinct points of the open upper half plane in $\Bbb C$. Using Orbit-Stabilizer, the Iwasawa decompositon $G=KAN$, and the Long Homotopy Exact Sequence, we can deduce:

Theorem. $\pi_1(\Theta_{r,s})$ is an extension of $\pi_1(G/T)$ (below) by either the braid group $B_s$ (if $r>0$) or its "alternating braid subgroup" (if $r=0$).

$$ \pi_1(G/T) \,\cong\, \begin{cases} \, \Bbb Z_2^{r-1} & s>0, r>0 \\ \, 0 & s>0, r=0 \\ \, \Bbb Z & s=0, r=2 \\ \, H_r & s=0, r>2 \end{cases} $$

Here $H_r$ is the subgroup of the spin group ${\rm Spin}(r)$ generated by $e_ie_j$ for $1\le i,j\le r$ (wrt a given orthonormal basis), subject to relations $e_i^2=1$ and $e_ie_j=-e_je_i$ ($i\ne j$). For example $H_3=Q_8$ is the quaternion group.

Theorem. The space $\Xi_n$ of all $n\times n$ simply diagonalizable complex matrices has $\pi_1(\Xi_n)=B_n$.

The space of $n\times n$ matrix solutions to $A^m=I$ is a representation variety whose components are orbits under conjugation. There is one orbit for each $n$-dim rep of $\Bbb Z_m$. Any rep can be encoded as a multiset of irreps (up to iso). The irreps of $\Bbb Z_m$ over both $\Bbb R$ and $\Bbb C$ are well-known. It is possible to translate "has dim $n$" and "has order $m$" each into a condition on irrep multiplicities.

Over $\Bbb C$, the orbits have fundamental group $\Bbb Z/g\Bbb Z$, where $g$ is the $\gcd$ of the dimensions of the irreps present in the given rep. Over $\Bbb R$, the fundamental group of the orbit is $\Bbb Z$ if $n=2$, else $\Bbb Z_2^{k-1}$ (where $k$ is the number of isoclasses of irreps in a given rep) if there is an odd-dim rep of multiplicity $>1$, else the aforementioned $H_k$ in the spin group.