This is a fairly common consideration in enterprise settings, to minimize help desk and IT overhead and to keep some level of order across My Sites (theme and branding particularly).
I've looked at this from a number of angles in past engagements and it looks like the system is making this task very difficult: Name changing back automatically and quota mails sent to site administrator are two of the biggest snags. You can get over them with proper code and procedure, but in the end the simplest solution we found was to create a permission mask on the MySite webapp and specifically block some rights from everyone except farm administrators.
From Central Administration you can specify permission masks that will apply to entire webapps, for specific groups. In your case, you could just block the "Set permissions" right, so they would simply not be able to fiddle with rights on that library.
Next you need to apply this mask to a group. Since it's a "Deny" mask you need to somehow make an AD group that includes "everyone but those allowed to set permissions", i.e. everyone but site administrators (if you apply the mask to just "Everyone", no one will be able to set permissions anywhere in the webapp).
We were lucky that our client already had in place automatic curation tools for their AD domain, so it was possible to add new users to a global group as soon as they appeared in the HR database. Administrator were excluded from that group.
Once created, the mask will be active and regular users will no longer see options to set permissions.