This thing has really been a love-hate... mostly because I'm learning as I go (hopefully I did everything right and it doesn't fall apart on me whilst whisking down the freeway).
I've put over 20,000 miles on my 400, it's a pretty reliable bike once it's set up correctly. Get the manual, it's not necessarily extremely obvious, but the info is there.
I can't speak to electronic ignition, but you definitely want a reliable battery. Most likely with the abuse yours has been through, you'll have to replace it. Going forward, keep it on the charger when not in use and away from large temperature changes should extend the life at least a little.
3. Had it running today, and it started leaking oil (a few drops) from the alternator. I popped the cover off and it's dry in there. Is this normal? Is there a seal that's missing? I was under the impression the alternator was oil-cooled but maybe I'm wrong and it's leaking from elsewhere?
I just replaced my alternator gasket, after starting the engine with the cover removed.. doh. Make sure the gasket surfaces are clean, relatively oil free when you replace the gasket. I used a razor blade to get some of the tougher stuff off, like old gasket. I ordered my gasket from these guys, they have two, I got the vesrah for 1USD more. 4into1.com/vesrah-alternator-cover-gasket-vy-1003-honda-cb350f-cb400f/
4. I rebuilt the carbs this winter, but now #3 seems to be leaking when the fuel is on and the bike isn't running... and tips I can try before I go ahead and take all the carbs off again and figure out what the issue is?
My first guess is float height. Unfortunately, you have to take the carbs off, especially for the interior carbs. I need to do the same thing. One temporary fix may be to use the center stand instead of the side stand so that the angle to leak isn't as great.
6. Finally - I haven't had my carbs synced or tuned yet, and I really have no idea where to start. The bike seems to idle at 2000rpm until it gets hot, and then it idles around 3000RPM... normal?
Both of those idles are high, start with a bench sync then move to the vacuum sync. I'm about to go through this process, just ordered a carbtune that should be here next week.
Love this bike... just wish I knew what it needed sometimes!
Patience, a nice full wallet, and tiny hands.