Infusion Product Groups sync to WooCommerce as Categories.   

Only Infusion Product Groups with active products set to Include in web are sent to WooCommerce, and the sync ‘cleans up’ any empty WooCommerce categories if there are no longer any active, Include in web products linked to the product group. 

Yes you can add upper (higher level) categories in WooCommerce and then link the categories that have sync'ed from Infusion to the parent categories you've added in WooCommerce. 

eg: This hierarchy will present correctly because the Infusion Product Group is the lowest level of hierarchy and contains the products. 

• Furniture  (this is a WooCommerce category)
  • Lounge Furniture (this is a WooCommerce category with parent loaded as  Furniture)
    • Sofas (this is an Infusion Product Group sent to WooCommerce as a category with parent loaded as Lounge Furniture). 
      • Infusion product 1 with product group = Sofas
      • Infusion product 2 with product group = Sofas

Yes there is a setting in Settings > Integrations > E-Commerce Settings:   

Send Product Groups

If this setting is unticked, Infusion Product Groups are not sent to the WooCommerce Back end and all product categorisations will be managed from the WooCommerce Dashboard.  

WooCommerce allows more than one category to be assigned to a product.  However if the Send Product Groups setting is ON in Infusion,  the WooCommerce categorisation will be over written when the product is next sync'ed from Infusion to WooCommerce. 

So -  Yes, a WooCommerce website can display a product in more than one category, but it will only remain there if Infusion Send Product Groups is OFF.