Unit group of octonions over finite fields One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: https://www.gap-system.org/Manuals/doc/ref/chap62.html .
When $K$ is a finite field, this is a finite dimensional $K$-algebra and thus has finitely many elements. Let $A_q$ denote the octonions over a field with $q$ elemetns.

Question 1: What is the number of units in $A_q$? Can one even describe the (possibly non-associative) group of units up to isomorphism?

For $q=2$ the order is 120 and for $q=3$ the order is 4320. In both cases it is indeed a group according to GAP.
Question 2 is motivated by Will Sawin's comment (I forgot the unit "group" might not be associative):

Question 2: For which $q$ is the unit "group" of $A_q$ associative?

It would be interesting to see what the smallest $q$ is such that the unit "group" is not associative.
 A: This is all worked out in the article "A class of simple Moufang loops" by L.J. Paige. The short answer is that the loop of units has size $q^3(q^4-1)(q-1)$, and is not associative for any $q$. The example given by Paige (lemma 3.5) is given in terms of Zorn vectors as
$$\left[\begin{pmatrix}
1 & (0,0,1)\\
(0,0,0) & 1 \\
\end{pmatrix}\begin{pmatrix}
1 & (1,0,0)\\
(0,0,0) & 1 \\
\end{pmatrix}\right]\begin{pmatrix}
0 & (0,1,0)\\
(0,-1,0) & 1 \\
\end{pmatrix}=\begin{pmatrix}
0 & (1,1,1)\\
(-1,-1,1) & 2 \\
\end{pmatrix}$$
and
$$\begin{pmatrix}
1 & (0,0,1)\\
(0,0,0) & 1 \\
\end{pmatrix}\left[\begin{pmatrix}
1 & (1,0,0)\\
(0,0,0) & 1 \\
\end{pmatrix}\begin{pmatrix}
0 & (0,1,0)\\
(0,-1,0) & 1 \\
\end{pmatrix}\right]=\begin{pmatrix}
1 & (1,1,1)\\
(-1,0,1) & 1 \\
\end{pmatrix}$$
so these two products cannot be equal over any characteristic. For $q=2$ we obtain the smallest simple nonassociative Moufang loop, which has order 120.
The article actually shows that a certain subloop modulo its center is a simple Moufang loop. At the time before Paige's result, the only simple Moufang loops known where the simple groups. Liebeck later proved the converse: Every finite simple Moufang loop that is not a group corresponds to such a subloop of octonions over some $\mathbb F_q$. In particular we shouldn't expect a simple classification.
