Can feeds importer grab fields, too?
It can, yes. We've used it extensively to import contacts (nodes) into a homebrew CRM.

After creating your Feeds Importer, you should see "Mapping" as the last item in your settings panel to the left. After configuring the rest of your importer to work with your export's format, the Mapping Settings of the processor will allow you to map source fields in your dataset to target fields in your actual destination content type. Your nodes should have the appropriate fields filled in on import.