Hidden Markov Model in Tagging Problem
Answer 1

You can try to directly fit the function $\Pr[y | x]$. This is called a discriminative classification. This is typically solved via some regression mechanism such as ordinary least squares, or lasso, or ridge depending on certain assumptions of the model.

Answer 2

The reason we want to factor $\Pr[x, y] = \Pr[x | y] \Pr[y]$ is because we are interested in the class conditional density $\Pr[x | y]$. That is, what is the probability of observing data $x$ given the label $y$?

There are various pros and cons to both discriminative and generative classification. For example, generative models tend to be easier to fit [naive Bayes is maximized simply by counting, whereas discriminative requires solving some convex optimization problem]. However, discriminative models typically faire better under independence assumptions when fitting data.