It is OK not to do all of these things for a given bug. Some bugs are trivial, or nontrivial to reproduce or write tests for. But you should explicitly decide not to do some of these things for any given bug, and be able to explain why.
Is there a (possibly old, closed) issue for the bug?
Can you reproduce the bug manually?
Can you write a regression test for the bug?
Did you check that your change actually fixes the problem?
Does the fix’s commit message explain the bug, the fix, and point to the issue?
It’s debatable whether checking for old or closed issues is the responsibility of the developer fixing the bug or a project manager who triages the backlog. Sometimes the second person doesn’t exist, but the job should still be done.
Remember that a bug report is a report of a symptom. A bug fix is directed at a cause. A software symptom may have more than one cause, making bug fixes that say "This fixes issue X" very optimistic about what they claim to achieve.
Exercise moderation and common sense in all things. Both 0% and 100% test coverage are awkward places to live in. Sometimes the information or effort needed to write a test make it too expensive to do so. However, when writing a test is cheap and easy or can be made so without causing harm, there should be a good reason not to do it.