Good luck with your trip. Flying with small children is a fair amount of work, need to be honest here, but EK does a good job helping out. EY, by comparison, are amateurs (unless the new air nanny makes any difference)
1. We're looking to take our car seat with us for the little one, but check it in as she'll be in the bassinet on the flights. Do we need to do anything with it, like package it in a box, or can it be checked in as is?
EK will typically provide a large heavy duty clear plastic bag at check-in to put the car seat or stroller in.
2. Has anyone taken the stroller with them to the gate? Do you get it back for transfer in DXB? Or should I expect to next see it in NCL in the oversized baggage?
EK allows you to take the stroller to the gate, but it will be checked at the plane and you will get it back at NCL. EK provides a fleet of standard Maclaren strollers at DXB for passengers (look for racks on the walls). These would not be appropriate, however, for an infant. We generally choose to check the stroller outright and use an infant carrier (e.g. baby bjorn, ergo) rather than the stroller in the airports. Easier to transit and keeps hands free.
3. Is there anything that we should be aware of that Emirates can/can't do in regards to travelling with an infant (I've heard they're one of the best in-flight)
Nothing comes to mind. You seem to already have booked a bassinet, so you should be set there. You probably know this, but don't forget to have the baby drinking something for take-off and landing to equalize the baby's ear pressure.
EK will provide some toys for the kids and a baby kit for infants (lotion, wipes, rattle) as well as baby food (but have never used it).
4. I have Qantas FF Gold - is there any chance I get lounge access anywhere with Emirates due to the alliance?
I don't know, but I'm sure someone can answer that here.
If not, you can always buy access to the Marhaba Lounge (small kids will be free)
https://www.marhabaservices.com/Source/lounge.aspx
Thanks in advance and I'll report back on how the flight was!
Best of luck on your trip