I am trying to transfer (or extract) attribute data from a polygon-feature class to point data. I can do this for a grid file using the Spatial Analyst-Extraction-Extract multi-values to points operation. But how do I do the same operation for a feature class layer? Or, alternatively, when I convert the feature to a grid file so I can use the operation listed above, my attribute table won't transfer to the grid table so I lose this information.
You could try using a region group (in the generalization toolset) to differentiate between different patches of equal value with an attribute table, that you could join with your polygon table attributes while still retaining the raster value. you might need to do a spatial join to match polygons to regions to make the connection work.
You can transfer attributes from polygons to existing points by using a Spatial Join and an Attribute Join.
If there are no matching ID fields (where each point feature contains an ID value for the matching Polygon feature) you need to first Spatially Join the two layers together. This will combine the attributes of both layers into one.
Once you have a layer that has matching ID fields, you can perform an Attribute Join to join either your polygon layer, or your new spatially joined layer, to your existing points layer. Then you can calculate fields to pull attributes onto your matching point feature. Alternatively you could just replace your point layer with your new spatially joined layer.