Can I upload files to my Pressero account and then link those files in on my product pages, pages on my site, et cetera?

Yes. You can upload any type of file to your Pressero account, and then link to that file on your sites. You have to know the path to that file and what address it will be available on. When you are working in your account, you can add files to your account from the Tools > File Manager command on the menu on the left of the admin area window.

If you are uploading an image to be available when customizing your site skin, those images would go into SkinImages. If you are adding any other types of files that you just want available or to link in on your site, use the Webfiles folder.
Once you have uploaded the file, and you want to find the path for that image, you can go to the file in the File Manager and click on the file, a detail window will be shown that will display the path to that file on the Pressero server.

The path is shown on the detail window for the file that you select:
The path shown on the file detail window does show the path to that file on Pressero. But Pressero has different installations and addresses that it runs on, depending on the type of account you have. Whatever root address you log into your Pressero account at will be the value that should come before the path shown.
So for the account shown in the above screenshot, that is an account on the US instance of Pressero, the value that should preceed the path would be, so the example image can be found at the address below.
For more detail on the different folders shown under the File Manager area, and more on how to use the File Manager, please see this article below.
Ch. 68 File Manager