Full Frame (FX), Crop sensor (DX) Confusion?

Hi, I am really confused on this. I know that the Crop sensor (DX because i use nikon) or APS-C i think it's called has a smaller sensor than a full frame. My understanding is that when you put an FX lens on a DX its focal length is multiplied by 1.5 (Am i right?), but when you put DX lens on an FX the image is cropped. When they say it gets cropped does it mean it actually creates a smaller image? So if a Full frame camera normally produced a 6000x4000 image would it be 4000x2666 or something with a DX lens on it? Please correct me on this if i am wrong.

So if an FX lens on a DX body is multiplied by 1.5 for focal length, would it produce the same quality? For example if i put an FX lens on a DX body with a multiplied focal length to create a focal length of 400, would a DX lens with a focal length of 400 be better quality? (Not including the quality differences in the lenses themselves!)

So to summarize if you put FX lens on DX body the focal length would be multiplied by 1.5 right? And if you put a DX lens on a FX body it would be a smaller image correct? Thanks!

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