JDW,
My experiences parallel yours in so many ways. I cast and play the rod/line/fish with my 'power' arm and control the line and reel with my left, but will swap the rod from right to left while fighting a large fish as the situation requires. I started when I was 11 and learned to tie flies then too. I've been tying almost all of my own flies ever since and have developed some patterns that are extremely effective for myself and a few friends. That's something that requires very smooth fingers too. I used 300 grit sandpaper on my fingers before each session.
Regarding which hand to hold the rod in and which to manage the reel and line with.... It's up to the individual...period!

Tradition is nice but sometimes it isn't comfortable.
There was a time when I studied each and every fly-tying book and entomology book I could get my hands on. I memorized all of the taxonomic names for all insects and every stage of their lives and when they occurred on the rivers and streams I fished. I guess I'm glad I went through that stage - the right gear and the right clothing too... These days you'll see me astream with a very old vest, a camo ball cap, no net, old patched neoprene waders, and three small fly boxes in my vest pockets.
I'll be that white-haired old guy who's having a good time without a care in the world who might sneak a sip from a small silver flask every whenever he releases a nice trout or salmon.