Is it possible to change the path of an image when it is uploaded via a custom field so that instead of appearing as www.mysite.com/wp-content/uploads/2014/05/file.pdf it will appear as www.mysite.com/page-name/file.pdf ??
Its not exactly clear on where you are going, and why you are going there? If you want to place the file.pdf at the location www.mysite.com/page-name/ go for it. Use your File Transfer Protocol (FTP) tool to create that subdirectory on your server, dump your file there, then refer to it via the proper url address.
If you want to override the wordpress tools that pushes media uploads at the uploads/year/month directory format, you will have to override wordpress core, not something for the faint of heart. I would assume you are very comfortable with php, and know your way around wordpress core functions in order to do that.
Is this a search engine optimization (SEO) issue? What's driving the desire for a change?