First, don't use regular crampons without a lot of training. There's a good chance you can make a minor slip or fall cause a very serious injury. Sure, it may help you slip less often, but I'd much rather slip and self arrest than to cut my leg open.
1. without training, is the axe still useful for crossing icy traverses?
No, not if you don't get any training at all. Read up and watch some Youtube videos. It's best if you get someone to help you train several of the falls, but I self arrested dozens of times this Spring without formal training.
2. how long/expensive is the training for proper use?
There are free courses. I know of one on South Lake Tahoe that's either 1 or 3 days.
3. once I took a class or learned from a qualified instructor, could I teach someone else to use the axe? The issue is that my traveling companion won't have time to take any training courses.
It's not like you get a certification when you take a class, and it's not like that'd mean anything anyway, so yeah, you could train your partner. I also advise the both of you learning how to self arrest with your trekking poles.
6. if I do buy an ice axe, is there a minimum length to make it useful?
There's a minimum length, but I haven't seen a commercial axe that'd be too short for me. I'm 5'9" and use a 55 cm axe. It's worked quite well for me.
7. if not, what's the lightest/cheapest ice axe that will still be reliable for traversing and self-arrest?
My ice axe weighs about 4.5 ounces. I've never wished it was heavier, although I certainly wouldn't have minded if it cost less.
8. for summer snow travel, will lightweight trail runners and light wool sock be enough to keep my feet warm while hiking?
If you keep moving quickly, yes.