You compile your Custom Shipping Provider to a new assembly with the CMS.Discoverable attribute. This allows Kentico to detect it and allows you to select the assembly on the Store Settings -> Shipping -> Carriers configuration page (note you also specify the class name on this page). Once this is done the Services will show up on the Store Settings -> Shipping -> Shipping options configuration page (ie. the GetServices function will be called on that screen).
The GetPrice function will also be called by Kentico when the relevant service is chosen during the checkout process.
As for calling the UPS Custom Integrated Class in the AppFolder, I would move this class into the Custom Shipping Provider. You would then be able to instantiate and reference it from within your shipping provider code. I wouldn't pass the deliveryitem class to the UPS class, instead use the routines that are already in that class and send in the necessary variable accordingly during the GetPrice method in your custom Shipping Provider.