E.E.
My son and I traveled alone to South Africa when he was 9 mo. old and have flown domestically a handful of times.
These would be my recommendations:
1)Take the travel system and carseat to the gate. Even if you didn't buy a seat for the baby, if there is an open one on the plane, you'll get priority and be able to bring you carseat on board. Though my son didn't stay in his seat the whole time, it was nice to have a safe place to lay him when he was asleep rather than having to hold him the whole. If you aren't able to take the seat on the plane they will either gate check it and you'll have it as soon as you get off. Just ask when you check in if the plane if full and the mention again that you have a baby when you get to the gate. There should be something marked on your ticket that you have a baby in tow and the flight attendants/ ticket counter agents will be extra helpful in securing you an extra seat if at all possible.
2)Be forewarned that you have to put the stroller through the x-ray machine at security. It's a bit difficult to fold up the stroller, hold the baby and take your shoes off at the same time so if you have someone to help you that's great. If not, definitely take some sort of sling or baby carrier with you. You won't be able to wear the baby through the x-ray but you can at least hold her while you fold up the stroller. Sometimes the security people are helpful and sometimes they stare at you like you have six-heads so plan for the worse. They make you take off the baby's shoes too at security so don't bother putting her in any until after you're gone through.
3)Though I packed a bag full of small toys for my son, he was more content to play with the airplane magazine and folding table than anything else. A straw kept him occupied for most of our flight home over the Atlantic so don't be too stressed about keeping her busy. Don't take toys you love because they are often lost under the seats and roll to the back of the plane never to be seen again.
4)Don't sit in the bulkhead, particularly if you're by yourself. Though the extra leg room is nice, not having ready access to your bag isn't worth the trouble. Blow-outs always seem to happen when you can't reach your stuff and it's difficult to keep an eye on the baby and reach into the overhead bin. Pack a carry-on/diaper bag of all the stuff she'll need (toys, bib, finger foods, diapers) plus a book/magazine for you.
5)If you have the trunk space in the car you'll have in LA, the bigger stroller, the better. Should the baby want to be carried in the airport rather than ride in the stroller, a big stroller can double as a luggage cart, nothing seems to fit right in the umbrella stroller. Plus if you're taking the carseat to the gate, the bigger stroller will make it easier to lug around.
6) Take lots of puffs and other finger foods and a good bib (like one that covers her whole body). If you have a fussy seat mate, the small snacks will keep her busy and quiet if necessary. The bib is vital- there will no doubt be food and/or drink spilled on her and you at some point during the trip. Pack a few changes of clothes for you and her in your carry-on.
7) Forget any worries you've had about germs and let her crawl, cruise, or how ever she gets around whenever she can. Bring wipes to wash off her hands after she's been roaming around but let her move as much as possible. You'll both go crazy if you try to keep her still and chances are she'll get a cold from the flight regardless of how much you try to stop it so just let her play.
8) Unless your daughter has a history of ear infections or screams during take-off and/ or landing, don't worry about feeding her or giving her something to suck on during those times. My son has never seemed affected by the change in pressure and I never fed him during those times unless it was his normal time to eat. If she's sleeping, don't wake her up to feed her because you're worried about it, she'll wake up if her ears hurt and the nap is much more valuable.
9) Though some people will recommend you give the baby Benadryl to help her sleep, unless you've tried it before on a plane and it worked, don't risk it. Kids often have the opposite reaction to the medicine, particularly in the air. I'd recommend taking some with you but more because if she had an allergic reaction to something on the plane you'd want it handy and there aren't a lot of drug stores in sky :) If on the flight there she seems to have trouble with her ears, you can give her Tylenol before take-off and about an hour before landing to help with the pain.
10) Enjoy your flight. I had a great time flying with my son. He's a very social little guy and made friends with everyone we met. The flight attendants and other passengers with remarkably helpful.