This is a mapping process. The distribution function is always the probability when the independent variable takes an interval, and then you determine the interval of the independent variable. Isn't that done?

As for why it became y, let's look at the equation transformation and read it several times. I feel that you haven't figured out the relationship between functions and independent variables in F(t) and G(m). T and m are just symbols, you can use them at will, but the function form is certain. You just need to substitute the independent variable into the corresponding functional equation and get the result naturally. I think your answer is very clear, in other words, that's it!