How can I renew my PR card from overseas without flying to Canada?
Not possible. You can only apply to renew if you are in Canada. Renewing takes about 4 months right now (or it did last time I checked).
And how long is a spouse who is a Canadian citizen a sponsor to the spouse from abroad? I divorced my husband and he threatened to drop his sponsorship telling me that if he does that I will lose my PR status because I don't live in Canada anymore.
He is lying. If you already landed as a PR, he can not cancel your PR status but you can lose it if you do not spend at least 2 years in Canada in every 5.
I am also accompanied by my child who is a registered Canadian citizen but does not have a passport at hand. Does that count as being with a citizen and would my days outside Canada count as if I were IN Canada???
No, being with a citizen child can not protect your PR because it is you who decide where you live, not the child. If you were the citizen and had a PR child with you, the PR of the child would be protected. The PR of a spouse is also protected if they are living with a Canadian citizen spouse.
How can I meet my PR obligations while in this situation??
Like steaky says, go to the embassy. Apply for a passport for your child and if your PR card is expired, apply for a travel document for yourself. Then go to Canada, apply to renew your PR card and make sure you stay long enough to meet the residency requirements.