Why does my Indian Ringneck bow its head and purr? Is it happy or sad?

This can be normal parrot behavior. They may flap and vocalize if happy, or communicating with thier "flock" which may be their human. They may also vocalize as mating behavior. If this doesn't answer your question please repost. Here's a little more info on normal bird behavior. https://lafeber.com/pet-birds/bird-behavior/

If Narashi is otherwise happy and healthy he may just be excited and playful and thus is flapping around and moving. If he seems in distress, can't balance, or is showing other signs of illness, then an issue such as illness, injury, neurologic or muscular issues, etc may be causing it. If you suspect this may be the case, an exam from your vet is best.

As chicks, one way to make feeding easier is to rub the side of the head in a circular motion. This will result in a yawn. Achu may be displaying this throwback behavior. It is likely no cause for concern. He may simply enjoy the massage and attention. Most birds enjoy being pet or rubbed on the head, neck, beak/nares and under the beak.

This is a lot going on! If she is getting scared near the cats, it is probably best to not bring her around them until she is more comfortable so that you don't get bitten and it doesn't stress her out even more. Making sure she is kept somewhere where the cats can't get to her cage and she can rest and relax is also a good idea. The nibbling or chewing sound could be a beak grinding sound, which can sometimes indicate the bird is attempting to rest and relax. If she's otherwise acting healthy and eating without bringing up food I wouldn't be worried unless it appears to be causing her stress or discomfort- in which case a vet visit is always best. If she's not yet comfortable with you, you can also start to gain trust by offering her a favorite treat every time you visit the cage. You may have to start with just tossing it in, then work up to handing it to her, and then petting and handling her from there. If she gets startled or is too nervous, take a break and try again later or go back to a previous step.

He may be regurgitating for you. It is characterized by an up and down pumping motion of the head as you described. This is normal. Mino is trying to get food/seed up to share with you. Consider it a high compliment from your feathered friend! As long as he is behaving normally otherwise (no lethargy, loss of appetite or loose droppings), there is likely no cause for concern.

By your description, it does not seem to be abnormal behavior. It may simply be something he does for attention. You can monitor for lethargy, loss of appetite, loose droppings or perched low with feathers fluffed. These are typical signs of illness.

She is likely screaming out of boredom. Birds are very social and require attention and engagement. Provide plenty of toys and puzzles. A lay area on top of her cage is also recommended where she can spend time out of her cage. When she screams, do not "reward" her by giving food. Ignore her until she is quiet then offer treats, individual attention and other rewards. Praise her and attention when she is quiet. Be patient and be consistent.