My husband and I have already received our Carte du Sejour (Passeport Talent) here in France which will expire on 2021. My employment contract with my French employer was not renewed so we're now going back to Philippines. Questions: 1. Can we enter to any Schengen country (other than France) from Philippines using our Carte du Sejour? 2. Will our residence permit be invalid once my French employment contract has been terminated?
Generally speaking, a carte de séjour does not become invalid when you lose your job (or interrupt your studies, etc.) The préfecture can move to invalidate it but this requires some paperwork (including sending a letter and giving you an opportunity to respond). Until then, nothing happens automatically. As far as I know, they typically don't bother, instead waiting for renewal time to catch people who do not fulfill the requirements anymore.
Furthermore, if you lost your job involuntarily, you're still entitled to renew the card for the duration of your employment benefits (that's two years if you worked at least two years, less otherwise). So even if the préfecture would check, as long as you are living in France, your card is still fully valid and/or can be renewed.
If you leave the country permanently and reside in the Philippines, that's different. You're not entitled to the unemployment benefits (and therefore to the card) anymore.