> I figured if sponsored person is inside Canada, then we only
> need Part A and Part B. (i.e. Part C is not required).
Sorry, you are incorrect. All part is needed. The application
will be sent exactly the same as if the sponsored child is
outside Canada. The sponsored child being in Canada does not
change that.
In this particular case, if you claim Canada as your residence,
the application will be processed in Buffalo, NY. So the region
you used for your application is "United States".
> But for people living in Quebec (and intend to continue to
> live in Quebec) they must submit additional forms and
> supporting documents.
That's true. For Quebec, please read here:
http://www.immigration-quebec.gouv.qc.ca/en/immigrate-settle/sponsors-sponsored/
http://www.immigration-quebec.gouv.qc.ca/en/immigrate-settle/sponsors-sponsored/
apply/
> If the parent have just moved to Canada and have not started
> employment, what other ways can a sponsor provide a proof of
> income?
The parent has income before, right? You can submit proof of
that previous income, e.g. proof of tax return in previous
country.
Note that if you only sponsor your child, there is no minimum
income requirement (unless your child already has a child of
his/her own).
> Would it be required to medical check again sine we already
> did the check one month ago in the country of origin?
If the medical check is indeed the same one, then AFAIK you
should be able to use it. However I am not sure whether the
medical check for TRP is the same as for PR. Did you bring any
form to medical? Did the doctor know it is for TRP not PR?

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Disclaimer: IANAL, IANALP, IANAMD, IANAMP, IANAAP
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