Honestly, a car seat is just one more thing to deal with - and you won't want that. Also, nix bringing a ton of toys - again, too much to deal with.
Bring a favorite snack, a loaded Ipod, a DVD viewer if you have one, a favorite lovey. It kind of depends on how long the flight is - but this was MORE than sufficient to bring my 3 year old on several 18 hour flights.
Airplanes are so exciting when they are little that they get less bored than you think they will..... Also, with long haul air travel, the movement and the noise tends to induce sleep.
Let her move around the plane, don't worry that she will annoy people (most people actually do like well behaved little ones....), and see if you cannot find another family to swap seats with if your flight is long haul (that way the kids amuse themselves, allowing the adults to rest)
The biggest stress in traveling alone with a toddler is not the plane - its the airport. Make sure to talk to her about airports being big places with lots of people, and the concept of "stick to me like glue" - do this both before getting to the airport, and before getting off the plane (the luggage claim area is fascinating to kids, they want to ride on the carousel.)
As to the stoller, it depends on how long the journey is. McClaren Volo is a relatively inexpensive, super strong, super light-weight stroller.... great for travel (and can serve as an extra luggage rack!) --- I traveled half the world with my son in it (meaning his naps didn't force me to stop my museum strolls) If you are going on a journey, you should have one.... if you are traveling to see friends and family, its probably not so necessary.