Answer to 1.
Unfortunately, you have to be a resident of Canada to sponsor your child. There are three requirements for sponsor eligibility:
a. You have to be above 18 years of age
b. You have to be either Canadian Citizen/PR holder (PR should be valid)
c. You need to be residing in Canada
The same conditions apply to Co-signer (which in your case is your spouse) of the application.
Answer to 2.
It really varies from application to application. Normally when you submit your application to CIC, you get AoR (Acknowledgement of Receipt) in a months time (by again this varies; some applicants have received AoR within a week's time frame as well). After you get your AoR, you eligibility of sponsorship will be evaluated (again the time frame for this varies - I have seen some users on this forum who got their sponsorship application approved relatively sooner than others). After the sponsor eligibility is evaluated and CIC accepts your sponsorship application, you will be notified for getting Medical Examination done for your child (if you are in US, you can get that done fast I guess). As your child is a minor, there wont be any background check necessary.
Answer 3.
In order to extend your child's stay in Canada as a visitor, you need to apply for visitor record (
https://www.canada.ca/en/immigration-refugees-citizenship/services/visit-canada/extend-stay.html).
However, I don't think that the sponsorship application period may go beyond 6 months period, especially when both parents and child are residing in Canada. If for any reason, the application processing takes more than 6 months, an easy way would be to enter US for three days and come back, this will give your child six more months.
Hope this helps.