The lens properties stay exactly the same. However, you might record the image differently.

Sensor size is going to be the biggest factor, as you noticed, as it affects both field of view and depth of field.

And since you're recording a smaller area in the center of the image circle, you're (probably — individual results may vary) getting the best part of the lens, and corner softness and possibly vignetting won't be as significant.

The smaller sensor doesn't inherently increase magnification. If you're printing the resulting image at the same size, that'll require you to magnify it more, but the size of the image projected by the lens remains the same.

The distance to the sensor is the focal length of the lens, and this doesn't change. Or it shouldn't — some combinations of camera mounts and lenses make adapters which put the lens in exactly the right place impossible, so focal length will change slightly and you'll probably lose the ability to focus at infinity (and possibly be able to focus a bit closer, which will increase magnification). This shouldn't be the case with Nikon lenses adapted to micro four thirds, though — the flange distance is so small that there's plenty of room for an adapter.

Other things, of course, will depend simply on the different properties of the sensor — depending on the camera, the photosites might be closer together, for example, and Nikon and Olympus are going to render colors somewhat differently. None of that is the lens itself, though.