Thank you for your post!
For your first question, the answer is no. In Aspose.BarCode, the HttpHandlers is configured for "BarcodeImageService.axd" rather than "*.axd", therefor it will NOT conflict with other application.
Why we use .axd extension is that, MS IIS were configured the association with .axd extension to Aspnet_isapi.dll by default. As we know, Aspnet_isapi.dll will finally send the request to your web application to process it.When IIS recive a request for "BarcodeImageService.axd", it will dispatch it to Aspnet_isapi.dll, and Aspnet_isapi.dll will send the request to the web application which was configured the handler for BarCode, so the web application will call the BarCode control to generate the barcode image and response to browser at last. If we used another extension (.abc etc.), the request will be block by IIS unless you reconfigure the association manually.
Trace.axd and precomplie.axd are two well-known application that using .axd extension, they are both design by microsoft. The first one is design for tracing the web application, the second is for precompling the application. These two application won't conflict with each other and, they won't conflict with BarcodeImageService.axd too. You can get more information about Trace.axd and precompile.axd from msdn:
Thanks again and please feel free to let us know if you have any question!