If I connect the traditional USB end of my cord into the old laptop and the USB-C end of my cord into the new one, will I successfully be able to transfer files from one device to another?
Depends on the laptops. I was able to do this with a pair of MacBooks. I booted the MacBook with USB-C into disk mode and then connected it to the MacBook with USB-A by a common USB-C to USB-A cable.
If I can do this, what will this look like, on-screen?
The MacBook with USB-A saw the MacBook with USB-C as an external USB drive.
The hardware for USB-C dual role ports are not unique to Apple hardware. Looking into the USB chips in most any laptop with USB-C ports will show that they use the same chips as Apple, and so should be capable of acting as a device with the right software. Even then this master/slave relationship not required for two computers to communicate by USB-C, a peer/peer system has been defined in the USB spec long ago.
Host to host communications by USB-A to USB-A cable has been in the USB spec since USB 3.0 in 2010. Use of USB-C on one computer, or both computers, isn't going to remove this functionality.
With USB-OTG in 2001 came the concept of a dual role device, and the dual role mini USB-AB port to make it work. USB-C brought dual role with more bandwidth in 2014, and this port became ubiquitous not long after. Oh, and micro USB-AB ports with the extra USB 3.x pins could bring dual role and gigabits per second too, I just didn't find the date that happened right away. It must have been introduced about the same time as USB 3.0. With a dual role port comes more options, like one computer pretending to be a USB drive. Apple took advantage of the dual role feature of USB-C but it seems nobody else has.
There is software to move data by USB from one computer to another but they will use a cable with an electronic lump in the middle. Well, it's not really a "cable" anymore with all the electronics in it, it's a two port USB device but it looks and acts like a cable to most people. These cables have become much faster and lower cost recently but the software I've seen for these cables are just horrible. The software I've seen looks like some FTP program from the 1990s. But then maybe I have not seen the latest options, and they have improved the user experience.
The electronics in these USB data transfer cables are all pretty much identical regardless of who's name is on it or where it was purchased, and some people wrote drivers in Linux to make them look like a standard network interface. If using Linux and one of these cables then it looks like an Ethernet connection, and people using them can transfer data with any of a number of means. That includes FTP programs from the 1990s.
There's documentation on how to set up USB ports for host to host communications with Windows and Linux but they are for outputting kernel debugging logs, not general purpose data transfer. To get software that allows for data transfer, and usable by the minimally skilled computer user, should only take someone sitting down to write it. Why isn't this a feature in every desktop OS by now? I have my theories but it would be nice to get a definitive answer.
I see questions like this come up often, and many people give the wrong answer on why it doesn't work. Any computer with ports that follow the USB 3.x spec has the hardware for USB host to host communications by a passive cable. This is a software problem, not a hardware problem.