Quantile regression is considered a natural extension of ordinary least squares. Instead of estimating the mean of the regressand for a given set of regressors, and instead of minimizing sum of squares, it estimates different values of the regressand across its distribution, and minimizes instead the absolute distances between observations.
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Simple regression is used when there is one independent variable. With more independent variables, multiple regression is required.
If the regression sum of squares is the explained sum of squares. That is, the sum of squares generated by the regression line. Then you would want the regression sum of squares to be as big as possible since, then the regression line would explain the dispersion of the data well. Alternatively, use the R^2 ratio, which is the ratio of the explained sum of squares to the total sum of squares. (which ranges from 0 to 1) and hence a large number (0.9) would be preferred to (0.2).
yes
linear regression
I believe it is linear regression.