Well, I've only ever traveled with one :-) but we flew many times from when he was 3 months old. I'd say go with the coach/seat for everyone option. I found it was much easier to strap my kid in, and have the extra space to stash the diaper bag, toys, etc. I know there is more room in 1st class, so maybe that's not an issue up there.
The other reason I always preferred to have the car seat is because it is much safer to be able to strap the baby down at take off, landing, and in case if turbulence. The odds of having any problem are small, but still...
We sometimes would just buy two seats (one for each parent) and got lucky enough to have an extra seat available for the baby. Flights are much fuller now, though, so I wouldn't count on that option.
Best of luck!
C.