Thanks Janet, I have had my eye on being right side dominate for awhile. I had personal success with that method in other arts...in Aikido I like my Left side Irimi Nage as my primary go to, where as on my right side I'll often use ikkyo or kotegaeshi.
I've heard some karate styles will focus on one hand more than the other in their kata. Most people are right handed, so the theory for some is train the strongest side much moreso to play to your strengths. Some want to create ambidextrous people, so the left side is trained twice as much as the right to correct a right dominant body. I prefer to be equally trained.
Also, as Uke I prefer to have equal bumps and bruises - I consider Kotegaeshi breakfalls to only one arm repeated impact to one hip and leg and lots of stretching to only one wrist and shoulder with no stretch or impact on the opposite side a recipe for injury for uke, for example.
Senior versus junior - usually I go first, especially if the junior as no idea what we are doing. I might be the one who has no idea, so I defer. Or, the beginner only started to get a feel after a few reps, so I let them go several more. Or, the teacher gives a correction to someone I let them go for a few times to work on the correction. Someone is testing and I've passed the test already, I'll let them throw much more often than me. I don't "pull rank" for katatedori dori Tenkan over someone who has trained for a year or so.
I did train in a school that liked to throw left, then right, then stop and talk, then your partner's turn, then walk around the room looking for someone else to do exactly once a side techniques with. I prefer to do more reps than that.
Really, the goal is to be healthy and to learn, so I don't let some arbitrary ettiquette get in the way of that.