Emma gave a few more pointers. Yes, try the "rename" tool and see what happens.
You do have a website for all of files, it's your server. I just gave a presentation at MoodleMoot2016 US that might be of interest to you somewhat about this topic. Here is my webpage for my resources, including my actual presentation (via video.) You see, this is a link to an external resource on my website.
For quiz questions that have embedding pictures, follow Emma's advice and see where Moodle thinks the picture is located (it's URL). The "rename" tool might then help.
Consider using underscores instead of spaces in your file names. Then the file name will look like "file_name" instead of "file%20name". This should bother anything on the Internet, I just find it easier.
I do not understand your backup system. You seem to be saying that you back up each file, file by file. The two backup methods that I know about are: 1) backup a course, one by one. 2) backup your entire moodle, the moodledata folder and the mySQL database, and even the moodle program directory to be safe. Do either of these two make any sense?
I don't have any thoughts about the differences in the picture URLs. Here is how one of my quiz picture's URL looks:
"http: //www.mydomain.com/moodle/draftfile.php/5/user/draft/937479655/ppg__pearson__tm__om11h__f19g1q12g1.gif"
Here is another idea. Backup on of your courses. Get a MoodleCloud account. Restore the course in your moodlecloud system. Remember the 200MB limitation and 50 user limit of Moodlecloud. If the 50 users is a problem, backup your course without user data. If the 200MB becomes a problem, well, you are out of luck.
Now that I am on this topic, I hate to say this. I have my courses on my "production" moodle. I have also successfully moved a course to Moodlecloud, pictures show. I have also moved my entire production moodle to my local MAMP experimental moodle system, pictures work. So your situation is puzzling since I have never ran into this problem myself.