EEA Residence Permit and Settlement Scheme

I am an unmarried Canadian partner of an EU citizen and just received an EEA Family Permit a few days ago. My partner is starting graduate school in October, but we will be moving to the UK at the beginning of September and I am close to potentially receiving a job offer. I have a few questions about the best way going forward:

- Can we apply for both the EEA Residence Permit and the Pre-Settled Status at the same time when we arrive in the UK? Which option will give me the quickest path to work rights?
- Can we apply for the EEA Residence Permit in September even though my partner's program starts in October?
- Even though I am unmarried, will the COA for my applications come with right to work given that I entered the country on an EEA Family Permit?