Why is my budgie whistling and keeping its wings slightly open?

Birds can hold their wings away from the body if they are warm, but it can also happen if they are having trouble breathing or are in pain. Since the other bird in your picture doesn't seem puffed up/panting as if it were warm, it may indicate that Totto isn't feeling well. If she shows any other signs of illness (loss of appetite, strange stools, lethargy), is having trouble breathing/panting heavily, or doesn't perk back up after a day it is best to seek veterinary care ASAP. Since you mentioned you aren't there daily, it is best to see if a worker at the retirement home can monitor her for changes/improvements. She may need to be separated from her cage mate while she recovers as well.

It's hard to see anything wrong from just photos however if you do have a video you can add in a YouTube link to your question for us to see. If Joy is holding the wing out funny, doesn't want to use it, or seems painful when doing so, it could indicate something such as an injury which should be checked out. Sometimes stress or activity can also increase breathing which may look like the wings shaking as birds expand and contract their entire chest-cavity to breathe. If she is otherwise normal and healthy and the wing is being used/isn't bothersome to her, I wouldn't be worried, but if she seems to have any problems breathing, the wing seems injured or painful, she seems listless, doesn't want to eat or drink, or shows any other signs of illness, having it looked at in person is always best.

If the area is very hot then your budgies could be very warm. Open mouthed breathing and holding the wings away from the body are both signs that a bird may be stressed from heat. If they are old enough to leave the box, you can try to mist them with some cool water from a spray bottle (spray above them, not directly at them) or place a fan in the room to help create a bit of a cross breeze until it cools down. Making sure the birds have plenty of access to water and a shaded area to rest in can also help them keep cooler.

Many birds, including budgies, will sleep with their heads nestled down over their backs and between their wings, especially at night. This is perfectly normal behavior. However, a bird that isn't feeling well will spend more time with its heads nestled down into its feathers and less time engaging in preening, eating, playing, and other normal activities. A sick bird may also stop vocalizing, sit puffed or fluffed up, sneeze or have discharge from their nares, have loose stools, have stools caked around the vent, and/or pick at its food or not eat at all. So, if your bird seems listless or is exhibiting any of the above symptoms, seek veterinary attention right away.

It would be hard to say without more context. Baby birds can click their beaks together if they're begging mom or dad for food. They may also just open their mouths to yawn and stretch. If the mouth is staying open and they are panting, it can indicate stress or overheating. If the babies are otherwise acting healthy it may just be normal behavior, however if you see any signs of illness such as lethargy, puffed feathers, refusing to eat, etc, then having them looked at by a vet is best.

Birds can sometimes do this motion to adjust their crop after eating, however if it is happening often, Warrior seems to be having trouble breathing, or there are other signs of illness (lethargic, not eating as much, listless, etc), a vet visit is best to rule out health issues.

The tail bobbing, opening and closing mouth may be indications of respiratory difficulties. This may bebdut ot infection (bacterial, fungal or parasites) or trauma. The fluffed feathers, when taken with other symptoms may further point to potential illness. Ideally, she should be physically examined by your veterinarian for diagnosis and treatment. Birds are quite adept at concealing illness or injury so by the time they start to show symptoms, they may be very sick.