Hi A.! What an adventure! LOL
I've done a bit of traveling (only 2 round trips with my daughter...at age 16 months, 18 months, and will be doing it again next month) 2 out of 3 on my own! It is REALLY an adventure! If you don't have a carseat at your destination (hmmm, maybe you can borrow one your friend might get early for her shower! LOL) that's the way to go...otherwise you can check it as baggage (I don't think they count it as a "bag" in terms of charges) or carry it on, same with a pack and play if you need it (I think at this age, an airmattress works fine in a pinch)...I usually bring an umbrella stroller...just makes navagating the airports easier (and keeps her in place while I check in or whatnot...and if she's not in it...my carry-on is!). You have to fold it up and put it thru the Xray maching though...that's always fun. You can use it right up to boarding and just check it at the gate. Be sure to put you and kiddo in slip on shoes if possible...you'll both have to take shoes off at security. Also, slip ons make it easier to take those shoes off when you get to your seat (sock-feet are less disturbing when they inevitably contact the seat in front of you, no matter how hard you try to keep them from kicking the seat!)
As for the flight itself...
I basically give up on all "rules" when it comes to sweets and videos! I bring a backpack filled with "smaller" bags...one for takeoff and landing (that I put behind me in the seat so I have it on hand all the time) with a couple small toys (preferrably new) to take out one at a time...a couple of "pushpop" Lollipops...smaller chance of choking, less messy and VERY valuable for take off, landings, and emergency "be quiet!" times. A bunch of small "post-it" notes in different colors...great to "decorate" the window and the tray table with (and easy to remove!). That bag is usually her "leash backpack"...she loves to wear it and I keep the "leash" tucked in the front pocket unless I absolutely need it (like when I'm juggling money, a bottle of applejuice and a fussy toddler!). Then another little bag with LOTS of snacks, one with a travel DVD player (yeah Elmo!) and headphones, another little bag with more toys, novelties, crayons, paper, stickers, and then the front pocket of the backpack with just a couple diapers, wipes, hand sanitizer, etc. and a side pocket for my cell phone and wallet.
This way, everything easily stashed under the seat in front of us for easy access and can be taken out as needed. Our first trip I took her on was over 3 hours and she was a "lap child"...that is the last time I'll do that! It is SO worth the money for a second seat if you have an active child. Just to be able to have the extra leg room (my daughter liked to stand up in front of the seat occasionally when she had her own seat) and a place to plop them while you get organized. Yes, they often spend a lot of time in your lap anyway, but it's much easier if you have a second seat/tray for snacks or DVD player. If she's going to be on your lap...make sure to not drink too much yourself before the flight and pee before you get on the plane! Those rest rooms don't give you much room to bring a kiddo in!
I bring an empty sippy cup and fill it once I go thru security with some juice so she'll drink for take off and landing. Also, I break my rules and let her have her "bink" (pacifier)during the trip as well (usually at home, it's bedtime/naptime only.
Being prepared can make YOU feel better...I usually end up still dealing with a crying, fussing, toddler, but least I know I tried my best!
Good luck!