The main impact should be that the group is no longer available for email distribution. As far as I am aware though, the inbound flow shouldn't be affected.
In Azure AD Connect, all three group types/scopes (Universal, Global, and Domain Local) can be synchronized, but there are limitations around the domain local and global scopes.
If you use group writeback, for instance, all groups are written back with the group scope of universal.
Where are you unable to select the distribution group? In order to synchronize the distribution group, you need to make sure that the following attributes are filled out: mail, displayName
If those are not filled, you may have issues with the synchronization.
If these are already set, please provide more details about where you are unable to select the group and provide additional screenshots, and I'll happily help troubleshoot.