We do not want to be separated when going to the passport line on arrival.
Go to the "all passports" queue. There's no queue that is limited to non-EU passports.
In fact, you can try going together to the EU-EEA-CH-only queue. You might get through and you might get sent to the other queue. I've experienced the former firsthand and I've heard stories of the latter.
Please correct me if I am wrong, I think the EU extends certain rights to non-EU family of a EU citizen where that family member is now no longer limited to 90 days in the Schengen, can go through the EU passport line, and possibly obtain the right to work/study. Is there a way to extend this to my unmarried partner?
You're not wrong about extending rights to non-EU family. Whether an unmarried partner counts as family, however, depends on the host country. Furthermore, durable relationship rights only exist after you have submitted evidence of the relationship that has been accepted by the host country. This is in contrast to the rights accorded to spouses and other close family members, which are automatic. In any event, it doesn't matter which desk you go to at passport control.
Does the EU have a specific definition of a "durable relationship" written into law and all the ways this can be proven? I just want to be sure in case it has since updated.
No. It's for each country to define according to its national law.
There used to be a EAA Family Permit, which I think no longer exists. Is there a modern version of this? My partner and I would be considered to be in a "durable relationship" according to the EAA Family Permit.
The EEA family permit was a UK document. It's no longer issued because the UK left the EU. The rules for establishing durable relationship for the EEA family permit were fairly clear because they were the rules established by the UK under its own law. They never applied to people seeking to establish a durable relationship for the purpose of EU free movement rights in any other country.