I've only scanned this thread quickly, so my reply might have been covered earlier, but here's my take on it. Mine is based on years and years of personal experience and five days of DAFNE course during which I learnt more than in the last thirty years, but I still have to adjust even from what that taught me, but it's those experiences that you asked for, so here goes. Please remember though, we are all different, and the super ball has edges on and doesn't always bounce the way you expect.
Of course, Aikido caught my eye. I did a year of it too: had to stop for many reasons but not because of my diabetes, and I miss it. It was a passion that I'd love to get back to. I also exercise a lot in other ways now, but I spent many years with very little exercise. Right now I have just started two different jobs: one with loads of walking and the other one cleaning. I found, like you, that my BS dropped like a stone when walking. I have always found that. As I was having repeated hypos with reduced short acting insulin, I have dropped my Lantus down. Very slowly, one unit at a time then leave it for a few days before doing it again. I'm getting there gradually and the hypos are getting under control again without too many highs.
I really wouldn't worry about a BS of 14 before exercise. I know the professionals say it's bad, but they (mostly) are not on the inside trying to juggle the eggs. They don't know how bad it is when said eggs break. You could try eating a few portions of slow release carbs immediately before your aikido class without any insulin. If your base insulin is right, it might be enough to keep the levels fairly steady during class without resorting to glucose. It will take some experimenting to find out how much you need to eat and whether or not to have a tiny bit of insulin with it, but it is something I do when exercising hard and it does work. I don't worry too much about testing unless I feel hypo, or 'off' in other ways, in which case the dreaded spike with exercise has probably happened. It does now and then, as you've discovered, and it sucks royally, but it won't do you any long term damage unless you stay up there a lot of the time. The odd spike is neither here nor there, long term risk wise. My personal theory, and I have not run this by any doctor, so they may say it's wrong, is that often during exercise that is stressful the body kicks adrenalin into the system (someone did say that above without saying 'adrenalin'). With Aikido if, for instance, you have a bad fall and hurt yourself, adrenalin will flood your body and hey presto, your BS goes sky high and doesn't come down. You cannot predict when such a thing might happen: you may have a busy but uneventful day or not.
When I ride in competition - I do endurance riding (horses) - I always have the will it? won't it? problem. It can depend on how wired my horse is before we start: if he's seriously wound up then I have to admit to worrying a little if the glue on my saddle is going to be strong enough, but sometimes the same can happen and I'm cool about it and so the BS comes down instead of goes up. More often than not when competing my BS tends to go up because Fella can be a bit on the scary side, so I carry my insulin with me as well as loads of juice. I tend not to use the insulin though unless the BS goes over about 17 because until it gets there the chances are it will come down on its own as I go further. With walking the opposite happens because there is no stress, just an hour or two of belting from door to door with catalogues, so the BS comes down. It's an eternal juggling act and the only way to work it out is to fiddle around until you find a system that works for you. And don't worry about starting exercise at 14: if your BS is still coming down then it's not a problem. At 17 you might find it won't budge and then it's a problem. Of course, those figures are the ones that work for me and may be very different for you.
When you are exercising hard, there is nothing wrong with starting a bit high: far better than dealing with repeated hypos, but don't let it stay high. Heck, I've said that already. Enjoy your Aikido and don't be afraid to talk to your sensei about the diabetes, I'm sure he (she?) won't bite.