What you're describing is not time travel. Those are just some of the simpler demonstrations of Special Relativity. In the case of the clocks for example, what happens is that time passed slower for the faster moving clock than for the clock left on the ground. Time travel would be if, for example, time for the clock on the plane had instantly changed from say 12:00 to 16:00 of the same day.
Wrong, one does not need to travel instantaneously to travel. Take moving a distance in 3D space. One does not jump instantly, that would be teleportation.
With time dilation under the effects of STR one travelling close to the speed of light would move forward faster in time and hence time travel into the future.
In fact we are all time travelling into the future now. If we weren't the present would remain the present and there would be no future.
Yes, time moves forward, but for each specific person their time is always moving at the same rate. For an inertial observer time in their own coordinate system always moves at the same rate, hence there is no time travelling