Hello! Hopefully I can help clear this up since both mrscruz and Lena1984 are correct, but for different specific situations.
In general, that is correct. However, for Mexico you cannot do that since the PGR background check requires an original, official request letter issued by a government entity (e.g., Embassy of Canada in Mexico). You can see the specifics for each country at http://www.cic.gc.ca/english/information/security/police-cert/north-america/mexico.asp
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How to obtain a police certificate - Mexico
Do I need to provide a police certificate when applying for permanent residence?
Yes, but not with your initial application.
The visa office where you submit your application will contact you when they need this information.
Correct, for the Mexico-specific situation and other very specific situations where an applicant cannot get the police certificate directly themselves. As you say further down in your post, you need to have an original formal request letter from the Embassy of Canada in Mexico (or another government entity) to get the PGR background check. In general, however, for family class applications the police certificate should be submitted upfront so if the applicant has lived anywhere other than Mexico those should be included in the application upfront.
It's because you can't get it any other way
Correct!
And on e-case updates:
My understanding is that, for privacy reasons, e-case is on a different system than GCMS and thus requires separate updating (the medical system is also separate, although that may be changing). If CAIPS (for older applications) or GCMS (for newer apps) is updated but not e-case then e-case doesn't reflect the actual status. Therefore, sometimes people are lucky and get a preview through ecase and other times they need to wait for e-mails or letters to arrive.
