Duplication and animation path.

I am not sure you are using the right word. If you duplicate a layer using the Edit/Duplicate menu or the keyboard shortcut Ctrl/Cmnd + d, every property you modified or added to the layer is on the new layer.

If you copy a motion path and then paste that motion path to a new layer you are only copying and pasting one property, position. If the original layer was set to auto orient along a path using the Layer/Transform/Auto Orient menu or the keyboard shortcut Alt/Option + Ctrl/Cmnd + o, that is a layer property, you cannot keyframe it, and it will not be applied to the new layer. Copying a motion path means copying two or more keyframes. When you copy, the keyframe values are copied, and the time between keyframes is copied, but the start time of the first keyframe is not part of the keyframe data that is saved to the clipboard. When you paste any keyframe data, the start time of the first keyframe value is always placed under the CTI (current time indicator) and if you think about it for a minute, this is the only thing that makes sense. 

If you have position and rotation keyframes on a layer you can select "Position" in the timeline, then Ctrl/Cmnd + select "Rotation" and then select the new layer and paste, both properties will be modified with the first keyframe in the set added under the CTI.

If you are just starting out, instead of digging through the UI or searching YouTube for interesting tutorials, please open the Learn workspace and spend about a half-hour learning the basics of the UI. Then spend some time with the User Guide and get a handle on the basics. After Effects is an incredibly complex application. Most of the new tutorials posted on YouTube are poorly explained and often inefficient recipes posted by enthusiasts. You have to vet your trainers to make sure they know what they are talking about. The number of views is no indication of the quality. Most of the folks having problems with tutorials are looking at pretty crummy tutorials. Premiere Pro and After Effects both require a fair amount of training to use efficiently, but you need to get a lot more training to stay out of trouble in AE.

Keyframe coordinates are always relative to the comp, not the layer. there is nothing wrong. If you really just want to rotate an animated object, you have to parent it to e.g. a Null and rotate the Null to change the coordinate space. I would suggest you actually read the help on such basics.

Mylenium