Committing a Name Mapping file
The extender file is actually one that should be excluded as it is a configuration file that is machine specific. You should make sure you have that in your .ignore.
As for the image files, they should not prevent execution if you ignore them. It's a feature of TestComplete that, when you map an object, it stores an image of the mapped object for reference. You can configure TC to NOT store those images.
Personal experience. The images don't do much as far as test reporting, especially when you're doing more data driven and dynamic testing. They take up a lot of space in source control.
If you need them though, there would be harm in keeping the image in source control.