In the benchmark hedonic price analysis, we assume a utility function of the general form
$$U = U(x, z_1,...,z_n)$$
where "$x$" stands for the composite good, and $(z_1,...,z_n)$ are the characteristics of good $y$ that are valued by the consumer. Assume for simplicity (as is usually done in the literature, and as is the OP case), that the consumer will only purchase just one unit of good $y$ (one house). The budget constraint in this case is
$$x+p(z_1,...,z_n) = M$$
where $M$ is the available income and $p(z_1,...,z_n)$ is the price of the good, expressed as a function of those same characteristics. This is an important departure from the standard consumer theory, because here, $p$ is not market price, but it is the "willingness to pay" of the consumer, decomposed into a function of many characteristics.
The lagrangean here is
$$\Lambda = U(x, z_1,...,z_n) + \lambda[M-x-p(z_1,...,z_n)]$$ and the f.o.c's are
$$\partial U / \partial x = \lambda$$
$$\partial U / \partial z_i = \lambda \partial p / \partial z_i, \;\;\; \forall i$$
So
$$\partial p / \partial z_i = \frac {\partial U / \partial z_i}{\partial U / \partial x}$$
Note that we have differentiated the price also, since again, this is not "the market price faced by the consumer" but his/her willingness-to-pay.
The hedonic-price econometric estimation is based on actual transactions, and so on data related to prices that consumers have actually agreed to pay.
So the specification
$$p = f(z_1,...,z_n, e)$$
estimates $$\partial p / \partial z_i$$ (and/or their binary-characteristic analogues), which is seen to be the ratio of the marginal utility of the characteristic scaled by the marginal utility of income.
If one assumes quasi-linear preferences of the form
$$U = v( z_1,...,z_n) +x$$
then $\lambda =1$ and so the marginal utility of the characteristic is the same as the marginal monetary willingness-to-pay for that characteristic.
On the basis of these remarks, I think you can conclude whether the econometric specification is a "reduced form" or not.
