I did my test 3 months ago, and was marked as completed a week later, and still have these three fields "in progress". I believe they start working on them after the test, but completing the test doesn't mean they will be marked completed immediately.
This is essentially correct. Language and presence will not be complete until at least the conclusion of the test (and ordinarily, but for temporary Covid related procedures, the interview), but presence and prohibitions are often not completed until some time after the test.
As previously noted, verification the applicant meets the language requirement is specifically done as part of the knowledge of Canada test and, again ordinarily, the interview. If the language element is not complete as of the conclusion of the test-interview step that means (usually, there are always some exceptions) the applicant failed to pass the language screening and will need to be retested.
Reminder: submission of proof of language ability with the application is only for the purpose of meeting the complete application requirements. That does not meet the applicant's burden of proving language ability sufficient to meet the eligibility requirements for the grant of citizenship. Applicants subject to the language requirement must still pass language ability screening no matter what form of proof was submitted with the application.
