First advice I would give you is start getting use to your bike by riding it as much as possible in a stress free environment (park, mup, etc). This way you will get the feel of the bike and starting/stopping and getting on/off will become second nature.
1. I don't see where you're from, but I do suspect that riding on the sidewalk is probably prohibited by the law, and even if it isn't, you will eventually need to ride on the pavement since getting on/off the sidewalks is going to get very tiring. In your case, I would start cycling on the street, but would chose to take the quieter ones with less traffic, even if it means adding more distance to your commute. As time passes and you become more acquainted with riding with traffic, you can venture onto streets with heavier traffic.
As for helmets; putting aside any laws that might be in effect in your area, I would definitely advise you on wearing one. It might not be the coolest thing to wear as fashion goes, but keep in mind that-that 250+grams of foam and plastic is designed to keep the most vital part of your anatomy safe!
2. I have the same problem as well, and try to balance as much as I can when I need to stop at lights. Whenever I can and it's safe to, I stop near the curb and rest my foot on the sidewalk while sitting on the saddle. Just keep in mind that-that's not the safest place to be since cars might try to squeeze past you thinking that you've stop to take a breather! When you can't use the curb, try sliding off the saddle, while straddling the top-bar and lean to one side so that your foot is resting flat on the ground.
3. I sometimes carry panniers (saddlebags on the rear rack) with items bulging out from both sides of the bike and additional items on the rear rack sitting taller than my saddle; and I'm what we call a Clyde (280lbs/ 125kg!). When my bike is loaded like that, I lean the bike over towards me as much as I can in order for my leg to clear the obstacles; in your case the basket, and then straddle the top-bar until I'm ready to roll. Back in the day when I used to ride motorcycles, it was easier for me to lean the bike (chopper back then) and pass my leg front - above the saddle (top-bar on a bicycle) since I couldn't swing it back to font because of the sissy-bar.
Starting off commuting on a bike is daunting, but as time passes and you get used to your surroundings, all of the above will become second nature.