1) As mentioned in Cody’s advice above, they like to shred and chew, not necessarily ingest. I’d just remove it anyway.
2) Buy a spray cleaner meant for birds. Spray and wait a few minutes, and it should loosen the debris to make it easier to get off the grate. There are different brands which come in a spray bottle. Some are basic cleaners made specifically to be safe for bird environments; some are disinfectant too. You can purchase this type of cleaner from specialty bird stores if you happen to have one in your area, or from an online bird shop.
3) It’s difficult to tell visually if a budgie has changed in weight. A gain is only obvious when they become obese, and the opposite, weight loss is even harder to tell. You may be seeing them when they slick down all their feathers, making them appear thinner?
4) Within a few days or couple weeks, you should see the pin feathers start to open up. I wouldn’t worry about disease unless you see the pinfeathers start to drop off or feathers not open up and develop properly. Don’t worry, your bird is just molting.
5) Only your vet can advise you on medicine and supplements. Make sure the injured bird isn’t being picked on by another.
If you haven’t already done so, please take s look at the Stickies in each forum section. They contain the most current information to help with useful tips and to answer some basic questions.
* Another thing.. you might look into getting some natural branches as perches for the cage. The uneven surface and diameter actually promotes foot health. Smooth dowel perches that come with cages ‘can’ cause pressure sores.