What is Box-Cox regression? I am confused about the terminology: What is Box-Cox regression? Is it applying Box-Cox power transformation and then running a linear regression? Is there any relationship between "Box-Cox regression" and "Cox model" in survival analysis?
 A: 
What is box-cox regression? Is it apply box-cox power transformation then run a linear regression?

It could be used to describe that but it will typically mean more than that. 
Consider that if you just look at $Y$ and find a Box-Cox transformation before you consider your $x$-variables, you're looking at the marginal distribution for $Y$, when the issue in regression is really (a) the shape of the relationships with those predictors and (b) its conditional distribution (especially getting things like conditional variance reasonably close to constant). As such you can't really hope to find a suitable transformation without doing it within the context of the regression itself.
So typically this would be "simultaneous" with the regression, not doing one thing then the other. For example, to use the MASS::boxcox function in R you pass it a model object. If you give it the same $y$ but a different model the estimate of $\lambda$ you end up with is different.
However, once you have an estimate of $\lambda$ in the context of a model, you can then transform your $y$ variable and rerun your model using regression (just as the routine to find suitable values of $\lambda$ does at each value of $\lambda$ it looks at).

Is there any relationship between "box-cox regression" and "the Cox model" in survival analysis?

No direct connection, outside the obvious one (Cox himself).
