Hi
1. If you have definitive proof from day one to show at least 365 days of cohabitation, then yes.
2. No, you can't apply early. Common-law is a legal status; it would be like applying as married a few months before actually getting married. If you apply with anything less than 365 days of continuous cohabitation, your app will be refused.
3. Anything that proves that you have been at the same address for at least a year. There are many threads on common-law proofs.
Unless your partner can qualify for another IEC permit or you guys get married and submit the sponsorship app by Jan so he can receive the OWP by May, he will have to stop working. There is no Implied Status as a worker for a person on an IEC permit.