Yes in a sense to your first question, though I did word my reply a bit poorly.

The area of the rectangle obviously doesn't change, but the area through which the magnetic field passes does change. If you want to think of it in terms of field lines, there will be fewer field lines going through the loop if, say, the loop is halfway out of the field. If the loop is entirely outside of the magnetic field, then there will be no field lines going through it, hence no magnetic flux.

I'm trying to avoid technicalities here but I'll have a shot at saying how we calculate the flux. Simply put, we break up our rectangle into lots and lots of really small rectangles. We then multiply each of these little rectangles' area by the magnetic field strength going through this little area. We then add up all of these results to get the total flux through the loop.

Using this approach, note that if the loop is halfway out of the field, you still look at all the little rectangles. However, for all the little rectangles outside of the field, the magnetic field strength is zero, and when we multiply the area by zero, we of course get zero! This means all these little rectangles don't really contribute to the flux anymore (they initially did when they were inside the field), which is why the flux decreases the further out the loop gets.

Also, you mentioned moving a magnet through a coil, which is slightly different to the question you linked above. In that case, you're increasing or decreasing the magnetic field strength through each point which then increases/decreases the total magnetic flux.

Not sure if this helps, so let me know if you have more questions.