yes. if v=c, we can say that the light particle isnt moving at all with respective to the moving clock, or its moving parallel to the bottom(or top) of the light clock without rising at all

the formula deals about one of the most mindboggling concepts ever, so holding nerve without trying to know how is it possible is a bit difficult but as we have so many other works, we tend to go on with the process and wait until you derive

The main objective of special relativity is what phenomenal changes occur when we move with tremendous speeds(i.e closer to light speed). So, u can’t expect an example closer to reality!

Yes, that’s what happens. Imagine the train is moving and when the origin of the train comes to the position of the x’, the same thing happens to the train origin as well. Dialation is happening from before, but we can we more clearly when train reaches that position, as you wanted to check both of the origins

not days but seasons as the earth moves in its orbit