In the line
$$\frac{d q_{2}}{d x}=\frac{\partial q_{2}}{\partial x}+\frac{\partial q_{2}}{\partial q_{1}} \frac{\partial q_{1}}{\partial x}= q_{1}+x \frac{\partial q_{1}}{\partial x}=\dots$$
In terms of dimensions of matrices, it looks to me like you have written
$$ [1\times n] = [1\times n] + [1\times n][n\times n] = [1\times n] + [n\times 1][n\times n] = \dots $$
because $x$ is a column vector. The product of matrices $[n\times 1][n\times n]$ is not defined, so there is an error here.
Let me try to rewrite in more explicit notation (I identify $\mathbb R^n$ with column vectors $\mathbb R^{n\times 1}$)
\begin{align} q&:\mathbb R^n \to \mathbb R,&q(x) &= x^TAx,\\
q_1&: \mathbb R^n \to \mathbb R^{1\times n}, & q_1(x) &= (A^Tx)^T = x^TA,\\
q_2&:\mathbb R^{1\times n} \times \mathbb R^n\to\mathbb R, & q_2(a,b) &= ab
\end{align}
Thus
$$q(x) = q_2(q_1(x),x)$$
Since there are subtly different spaces involved here, I'll elect to use a notation that works for arbitrary spaces, namely the Frechet derivative $d$ which always satisfies the chain rule
$$ d(f\circ g)(x)[h] = df(g(x))[dg(x)[h]],$$
and then the only game is to figure out what spaces each differential belongs to.
Letting $\partial$ denote a partial Frechet derivative, we have
$$ dq(x)[h]= \partial_aq_2(q_1(x),x)[dq_1(x)[h]] + \partial_b q_2(q_1(x),x)[h].$$
Now, $q_1$ is linear, and $q_2$ is separately linear in each variable. For any linear map $L$, the Frechet derivative $dL(x)[h]=L(h)$. (The nice thing is that this covers multiplication of matrices on the left and on the right.) Thus for $h\in\mathbb R^n$ and $v\in\mathbb R^{1\times n}$,
\begin{align}
dq_1(x)[h]&=q_1(h)=h^TA,\\
\partial_a q_2(a,b)[v]&= vb,\\
\partial_b q_2(a,b)[h]&= ah.
\end{align}
therefore \begin{align}
dq(x)h&=dq_1(x)[h]x + q_1(x)h\\
&=h^T Ax + x^TA h \\
&= x^TA^Th +x^TAh \\
&= x^T(A^T+A)h = ((A^T+A)x)\cdot h
\end{align}
and finally we can read the Jacobian from the final line:
$$ \frac {dq}{dx}(x) = x^T(A^T+A).$$
(And, the gradient vector is its transpose $\nabla q(x) = (A^T+A)x$.)
Equivalently, one has to understand that the Jacobian of a map $\mathbb R^n \to \mathbb R^{1\times n}$ acts on the right: if $g$ is $\mathbb R^{1\times n}$-valued, then
$ f(x+h) \approx f(x) + \frac{df}{dx}(x)h $ cannot make sense by dimension counting (remember $h$ is a column vector). So we are forced to accept that the correct approximation is $f(x+h)\approx f(x) + h^T \frac{df}{dx}(x)$. What this means is that the order of terms in chain rule is reversed! Indeed, the standard chain rule formula for the composition of a $\mathbb R^{n\times 1}$-vector valued function $f=f(y)$ and $g=g(x)$ is
$$ \frac{d(f\circ g)}{dx}(x) = \frac{df}{dy}(g(x))\frac{dg}{dx}(x)$$
If we take the matrix transpose of this equation, we obtain the chain rule for row-vector-valued functions $h=f^T$:
$$\frac{d(h\circ g)}{dx}(x) = \left(\frac{dg}{dx}(x)\right)^T\frac{dh}{dy}(g(x)).$$
A similar issue occurs in the above where the input is the row vector, and the approximation for $q_2$ reads
$ q_2(a+v,b)\approx q_2(a,b) + v \frac{\partial q_2}{\partial a}(a,b)$. Hence (and I think deriving from first principles is easiest)
\begin{align} q_2(q_1(x+h),b)&\approx q_2(q_1(x),b) + (q_1(x+h)-q_1(x)) \frac{\partial q_2}{\partial a}(q_1(x),b)
\\&\approx q_2(q_1(x),b) + h^T\frac{dq_1}{dx}(x) \frac{\partial q_2}{\partial a}(q_1(x),b)
\\& = q_2(a,b) + \frac{\partial q_2^T}{\partial a}(q_1(x),b)\frac{dq_1^T}{dx}(x)h \end{align} The correct chain rule is therefore, as painful as it seems,
$$\frac{dq}{dx}(x)= \frac{\partial q_2^T}{\partial a}(q_1(x),x)\frac{dq_1^T}{dx}(x) + \frac{\partial q_2}{\partial b}(q_1(x),x)$$
It would be easier IMO to avoid this mess by not using a $\mathbb R^{1\times n}$-valued function.
