Well, as more than 48 hours passed between my MacBook Air becoming completely unusable and your reply appearing here, naturally I had to look for help elsewhere.
I am replying for the sake of others who might find this discussion in a similarly dark hour.
I tried all the things you suggested, thanks to other sites and forums that gave me advice. Nothing worked. Even "Safe mode" did not work. I also tried booting in Recovery mode. That opened the Disk Repair Utility. After "verifying" the disk failed, I tried to repair the disk. The disk could not be repaired. I feared my data was all lost.
The next morning I spent an hour or two on the phone with Apple Support. The support person had me do all the things I had already done. In the end, he said I would need to erase and reformat my hard disk, thus erasing all my data.
https://github.com/macloo/recovering-mac-files
That is a step-by-step list of how I recovered my data. No thanks to Apple Support.
Remember, this all happened because I did a normal Apple Software Update in the normal way. When I clicked restart, the computer was never able to restart. I keep my MacBook Air very well up-to-date, usually installing everything within a week of the update's release. I won't be doing THAT in the future, as clearly this update was seriously flawed. It's clear that a lot of people had this 50%-stuck-on-reboot error. You can search around and see that. Whatever software update Apple released in early March, it was a disaster.
I'm lucky. I paid for a 3-year Apple Care plan. When I took my Air into the authorized Apple repair place and they ran a diagnostic on it, they confirmed that the solid-state hard drive in my 2013 MacBook Air had bad sectors and cannot be repaired. I say I am lucky because: (a) I had recovered all my data, and (b) the new hard disk and installation costs are covered by Apple Care.
Of course, I lost hours of time, and I have to be without the Air for five days at least (weekend), and when I get it back, I'll have to reinstall all my software.
The moral of the story is: Do not trust the Apple updates. Back up all data before doing any kind of Apple update, and ideally you should wait and then Google around and see whether the update is killing people's hard disks before you install it.