1. I am aiming to get travel sizes of my toiletries because I'm only taking a carryon bag and not checking luggage. I will be staying at a hotel, but can you tell me if I'm forgetting anything?
contact lens solution, conditioner, moisturizer, and hair styling stuff (I'll squirt some in a tiny bottle?). I am relying on the hotel to provide hand soap, shower soap and shampoo.
Toothpaste.
My carry-on kit has travel sized toothpaste, mouthwash, moisturizer, facial scrub, and shampoo. When I travel with a young child I also have a bottle of kids tylenol on hand. I never carry soap as I find it to be widely available

Bed Bath & Beyond has a great selection of travel-size items, btw. Pack all the liquid items into a single quart sized ziploc and pack it somewhere accessible so you can pull it out at security screening. You child can either pack her own items and pull them out (if she is old enough); you can pull them out for her; or if you can fit them all together, they can all go in your ziploc.
3. I am really scared of getting lost at LAX and not being able to find Super Shuttle. Help me to not cry.
You'll need to figure which terminal you are flying into. Each airline gets its own bank of gates, so you figure your terminal by cross checking which airline you are flying:
http://www.lawa.org/LAXAirlines.aspx
Here are maps of LAX terminals.
http://www.lawa.org/welcome_LAX.aspx?id=256
6. If I am picked for a random extra screening will they let me keep my child with me? I am very scared of losing her in a big crowd at the airport.
They will. How old? If the child is old enough to just wait for you, that should be no problem at all. If your kid is younger, you will be allowed to carry them through the screener. If you're selected for secondary random screening, tell them that you are traveling alone with a child and that you need a safe space to put her down. If your kid is a 2 year old who is liable to bolt given the opportunity, you might ask if you can be screened in a private room, with your child present. I'll note that you probably arent' terribly likely to get a random screening. Anecdotal evidence suggests that wearing a skirt increases the likelihood.
Other things to know/random tips:
I bring empty water bottles and fill them at water fountains after security. You can find water fountains outside the restrooms.
Bring food on the flight with you. The purchase options for in-flight food are expensive and lousy.
As well as shoes, I generally take off my belt, because the buckle makes the metal detector beep (but not all the scanners are the metal detector type any more).
You'll be fine!! Have fun.