<p>I just assembled a Prusa i3 MK3 and went through the calibration process, but when I print the first layer doesn't look good and my prints come unstuck from the bed. I think it might be Z height but this was as high as I could put the probe without the paper moving on the calibration test. The layers after the first few look good but then the print either warps or comes unstuck and moves.</p>

<p><a href="https://i.stack.imgur.com/EdidQ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/EdidQ.jpg" alt="Bottom view of a hexagon print"></a>
<a href="https://i.stack.imgur.com/4voqC.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4voqC.jpg" alt="Detail of the bottom of another print"></a></p>

Judging by the images you posted in your question, the first layer distance is too far away from the bed for the current filament flow.
This could either be related to:
having an offset on the first layer like a height correction in the slicer,
an incorrectly levelled (read height adjusted) bed, (you did the paper test correctly, so this is probably not your problem, it is mentioned for completeness)
under-extrusion
slicer setting not correct, e.g. filament diameter or flow modifier not 100 %
incorrectly calibrated extruder
Your most likely problem is under-extrusion. It would be advised to calibrate the extruder:
How do I calibrate the extruder of my printer?
and check the slicer settings.