We have had a car seat bag from Amazon. It even says “baby car seat” on the bag. You could also put on your back if your hands were full carrying cases, the trolley etc. I personally wouldn’t bother with a box, as it’s more difficult to handle without the straps etc.

Like many others, we padded it with nappies/coats/towels and even baby food pouches on one occasion. We have never had any issues with the condition of it on arrival.

As for where to take the bag at the airport. Take it with the rest of your luggage then they will direct you to oversize baggage. I always politely ask if they can tag it as fragile before.

I will mention that we have had issues with it arriving before on BA once when we flew Manchester via Heathrow to Singapore. It did arrive in the end though. Oh, and then 3 days later in Singapore we flew Jetstar and they asked to put it on a trolley close by and it never arrived in Bali. I felt like the car seat was chasing us around Asia. I do think I may just be a tad unlucky because last weekend our stroller didn’t turn up in Prague. Never no damage though.