About the maximum likelihood, when we convert the maximization problem into minimization, why we take the negative?
It is common to define optimization problems as minimization problems instead of maximization. And by multiplying your target functions with $-1$ you can transform one into the other:

$$\max_{w} \log{L(w)} \Leftrightarrow \min_{w} -log{L(w)}$$

So to maximize the log-likelihood you minimize the negative log-likelihood. Basically it just comes down to conventions in optimization theory. 

Moreover, since $L(w) \in [0,1]$ its logarithm $\log{L(w)}$ will be less than or equal to $0$ (note that $log{0}$ is not defined). Accordingly $\max_{w} \log{L(w)}$ means to maximize a negative number which is, at least to me, less intuitive than minimizing a positive number.  

The more interesting part is actually the log-transformation which increases numerical stability of your calculations (since it "transforms" the multiplication to a sum and thereby reduces the risk of underflowing).