I don't think it's a bad habit to not have a dominant leg for clipping out/putting down on the ground when stopping. Having a dominant leg is not an intentional thing, it's just the way you move instinctively, like being right-handed or left-handed.
For D, I suggest what Crankin is saying except that I start with the clipped-in foot higher up, at 2:00 rather than 3:00. That gives you a bit more momentum as you sit on the saddle and clip in the other foot. The important thing is to be seated right away because you're more stable that way and less likely to have problems if you don't get the second foot clipped in the first try.
I also agree on practicing in a parking lot, or on an unpaved field. What you want is to develop muscle memory so you can clip in and out without having to think about it.
I don't know what to recommend for the irrational fear of falling over. Though I think it might not be as overwhelming as you think -- you're staying on the saddle and balancing on your tiptoes when stopped even though you admit it's not stable. I think you're more likely to fall over than you'd be if you unclipped one foot, got off the saddle, put that foot solidly and the ground and stood over the top tube.
As for the people who are making fun of you, who cares what they think? You've managed to rack up a few thousand miles so far, no thanks to them. Just enjoy the ride, that's what matters.
- Gray 2010 carbon WSD road bike, Rivet Independence saddle
- Red hardtail 26" aluminum mountain bike, Bontrager Evoke WSD saddle
- Royal blue 2018 aluminum gravel bike, Rivet Pearl saddle
Gone but not forgotten:
- Silver 2003 aluminum road bike
- Two awesome worn out Juliana saddles