I'm only 5'2" so I know what you're going through. You will probably go through a period of getting a lot of advice from people, but the reality is, unless they are your height they don't really understand the challenges you are dealing with (despite their best intentions). So listen to what people are offering, and consider the rationale behind it, but find what works best for you.
My first bike, a CBR250RR has a seat height of 725mm, so perhaps was a little easier than the new ones with a seat height of 780mm - even still I remember being incredibly nervous that I couldn't 'flat foot' both sides.
Firstly, braking. There was some advice to given earlier which IMO is pretty poor. When doing slow manoeuvering i.e. in traffic, you should use your rear brake as it will be smoother for you and the bike. However, if you are coming up to a normal stop then you should use your front brake first and then apply your rear brake as well. This is the most effective method of braking and, importantly, is the method you should use for emergency braking. If you get into the habit of using the correct braking method all the time, when you have to actually do an emergency brake it will occur without you having to think about it.
When you come to a normal stop, road surface and camber is far more important to us short riders than anything else. You should be able to come to a stop and balance briefly before you need to put your leg down - something that will come with a bit of confidence and practice - so don't be too concerned about which leg you 'should' put down. I nearly always have my right leg down because I'm right-handed and my right leg feels marginally stronger and safer to hold the bike.
But, as I said, being a short rider, you have to check the road surface and camber before making that decision. If the road is not flat then put down the leg that is closest to the ground. If you come up to a set of lights where trucks and heavy vehicles have left dents in the road from their weight, then always make sure you stop in the actual groove/dent. If you stop on the high point, you're making it harder to get your feet down.
Stopping at lights - it sounds like you might be taught a different theory in US than in Aus, but I was always taught to keep the bike in gear and clutch in at the lights. The rationale is that if some idiot comes up behind you and fails to stop in time, you have a better chance of getting out of their way if you are in gear and can take off quickly. (Also, in Australia - if you have filtered to the front of the lights and you don't take off quickly then you will have some irate moron up your backside pretty quickly)
Importantly, when you're coming to a stop make sure you keep your head up. I know it's one of those basic things, but it will really help with your balance when you are stopping - and balance for us shorties is pretty important.
You're friend noticed you putting your foot down a little early (and I noticed it was the right one too!) I wouldn't worry about this too much - you will do it less as you pick up a bit more confidence. But this does mean you're currently not using your rear brake when coming to a stop (see point above about correct braking method).
Hill starts - yes, you're meant to use your rear brake, but if the hill has a camber on it as well then that won't always be an option for you. Practice hill starts (not that it looked very 'hilly' where you were!) by feeling where the take up point is on your clutch and having only the front brake on (because for some hills you are going to have to have your right leg down and won't be able to use your rear brake)
Finally - something that no-one will tell you........... I've found the best thing to do for us shorties before hopping on a bike is to do some hamstring stretches. You will be amazed the difference it makes and how much 'longer' your legs seem. If you wear heels a lot (like me) I find that it is essential for me to stretch before I hop on my Ducati.
Oh - and get yourself some kevlar jeans and some proper boots! Alpinestars make small sizes for women (if you also have small feet). My SMX4s are a European size 35.