A few tip-overs are a pretty common part of the learning curve ... they sure were for me. Usually the worst bruise is to the ego. Hope that's all you've had.

+1 on making sure the pedals are set pretty loose while you're learning. Did your fitter loosen them all the way?

I didn't quite understand what you said about tipping over to the left even after your right foot is on the ground, though. Are you trying to stay in the saddle while you're stopped? Most bikes aren't built to let you do that. Get in the habit of coming forward off the saddle as you bring your foot to the ground, and hoisting yourself back up as a part of powering up to take off.

Basically, clipped or not, you usually want to keep a foot on a pedal when you're at a stop sign or stop light, to let you take off as quickly as possible when you want or need to. There's no reason to put both feet down every time you stop, and safety/maneuverability is a reason not to.

You'll hear different opinions on whether you should make a habit of always unclipping the same side. My feeling on it is that getting used to one side will let you build muscle memory more quickly. Once you're very confident getting in and out of the pedals, then you can practice on the other side - it's good to be able to unclip either side in case you need to for whatever reason.