Creating a grid with inherited attributes from underlying features

I have a series of features with various values, and a vector grid overlaid on top of the features. I need each square in the grid to inherit the value of the underlying feature. If the square happens to fall over two features, I need that square to inherit the attributes from the feature that covers the square the most. Finally, if a square doesn't cover any feature, I need it to inherit the values from its closest feature.

I have tested out various methods so far and they don't seem to work very well. If I convert things to a raster and use euclidean allocation, I start to get a result that I am looking for, but then the rasters create alignment issues and I lose a lot of accuracy. If I can keep these in vector format that would be much better.

Any suggestions would be greatly appreciated! Attachment 26330

Thank you.

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