The answer is no, your likelihood is not correct because $\mathbf{x}$ is a random variable. Equation 2 should be:
$$
p( \mathbf{y} \mid \mathbf{x} ) \quad \text{not}\quad p(\mathbf{x})
$$
You have a prior distribution on $\mathbf{x}$. MLE is suitable for "deterministic but unknown" $\mathbf{x}$. Also, you observe $\mathbf{y}$ not $\mathbf{x}$ (or at least that is implied by the use of $\sigma_v$).
You can approach your problem using conjugate priors (Bayesian, as Mark Leeds suggests)
https://en.wikipedia.org/wiki/Conjugate_prior
or as a MAP estimate
https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation
which is better is application dependent. You need to choose. The MAP handles range constraints on unknowns nicely, it is often a hybrid MLE.
The Normal or Gaussian distribution has a conjugate distribution (although knowing or not knowing the variance on the prior is different) so you have a posterior pdf as your estimate, at which point you can take a mean, or a max, or a median as a point estimate.
With the advent of Monte Carlo Markov Chain (MCMC) and programs like BUGS and Stan, the Bayesian approach is gaining ground.
https://en.wikipedia.org/wiki/OpenBUGS
https://en.wikipedia.org/wiki/Stan_(software)
One of the best things about being a Signal Processor, is that often, different sets of assumptions will produce acceptable algorithms, but now you can trade off complexity versus throughput, hardware, battery life, cost ....
Statisticians tend to be more particular about how assumptions relate to bias.