We are investigating the possibility to change the slug/name a Sentry project. Does changing the slug/name of a project affect the project's DSN? We do not want to affect current live reporting by changing it.

Renaming a project will not affect the DSNs or the existing implementations. The only detail you need to be aware of is that any implementation that uses the project slug/name will need to be renamed, such as in sentry-cli and source map/debug file upload setups.