If you leave the earth’s atmosphere (60miles, debatable) and you come straight back down, would you always land in the same place?

No.  Ignoring things like wind and air currents that would move you from your starting position, the difference in circumference around the central point of rotation when maintaining the same lateral speed would cause your position to change.  At the surface you have some specific lateral movement (your angular momentum) due to your position on the rotating planet.  When you go up, your distance from the axis of rotation gets bigger, meaning the circumference around that axis gets proportionally bigger.  You would have to go faster to maintain your position over your starting point at the surface, which has a smaller circumference to travel around as the planet rotates.  As you go up you don't have sufficient angular momentum, and so you would fall behind as the surface of the Earth starts getting ahead of you.  As you come back down your circumference shrinks back down to match your lateral speed of rotation around the Earth's axis, and theoretically you would have zero horizontal speed when you touch back down.

This phenomenon can be observed by dropping an object down a long vertical shaft like a well, drilling pipe, etc.  As the object drops, it has more lateral speed than it would need to maintain the same rate of rotation or orbit around the axis of the Earth's rotation as its circumference around that axis shrinks, and that results in the object moving sideways--in the direction of the Earth's rotation--until it hits the side of the shaft.

Ok, the earth rotates 1000mph, so if you leave the earth’s atmosphere, wait an hour and then come back down, will you end up 1000miles from your original location?

No.  The Earth is rotating at approximately 1000 mph (3,600,000 mile per second).  A object in free fall accelerates at a rate of 32 feet per second per second (minus air resistance that increases as you get closer to the surface).  I don't have the desire to work out the math but suffice it to say you will come down a few miles from where you started.

That would both depend on your velocity going up, and the arc following engine shutdown, and the lift created by whatever vehicle you would be in.

Generally speaking no. The Earth's rotation and atmospheric conditions would make it quite difficult without some means of mitigating them.

Yes, in a nude beach...