For the record, we have a ticket about the missing synchronization of issue type, but it has never received a lot of traction: https://jira.sonarsource.com/browse/SLE-167
AFAIK changing issue type is not very common, and I think it is more likely to be removed from SonarQube than added to SonarLint.
If you are really interested by the feature, I suggest you open a new thread in the category “Suggest New Feature”, so that we could start collecting votes and use cases.
Yes, this is the SonarLint icon.