All these lines vary slightly from bird to bird. Photograph any unusual physical features, like scars or marks or missing toes and the like. You can get the bird tattooed with an ID number, but bird skin is thin, and doesn't hold the tattoo well. It fades quickly and has to be retouched every so often. It also is very stressing for the bird.
Transponders are effective, painless and undetectable.
Yes, there is. You may be able to get additional coverage from your company, but usually not. However, there is a company called Avi-Care that has a standard policy which provides coverage against theft and death. There is also an optional veterinary coverage at additional cost.