Why is my budgie regurgitating seeds? Is he sick or just bonding?

This may be mating or sexual behavior. The whole seed indicates regurgitation instead of vomiting. The rubbing of the perch is likely mounting behavior. It should pass. You can provide different toys or rearrange the cage to distract him, but there is no cause for concern as long as he is behaving normally otherwise.

Birds do not recognize themselves in the mirror, so this would explain the mating behavior of regurgitating. I know it looks somewhat concerning, but it is more of a behavioral thing and is not concerning from a medical standpoint. If she seems more hormonal (like she's displaying a lot of mating behaviors), then you should expect this to pass within a few weeks. Unfortunately, there is not much you can do to stop her from doing this at this time...you'll have to just wait for this phase to pass. If she doesn't seem to be keeping any food down or is continuing the behavior for longer than a month, then it's best to have a vet check her out. There are a few different infections and conditions that cause chronic regurgitation, so it would be good to rule these out as a cause.

Birds can sometimes begin to regurgitate or share food if they're feeling in a "lovey" or mating mood. It may be that he is considering you a potential mate. It'd be best not to encourage the behavior if this is the case, and to try and interact with him in other ways, such as offering a toy to play with, walking away when he attempts it etc. If you're seeing any other symptoms such as changes in eating habits, stool changes, listless behavior, etc, then it could indicate something beyond just behavioral, and scheduling a vet checkup is always a good idea.

I would move A Rod to a separate hospital cage as if he's sick being chased will only stress him out more. It could be the stress of the female causing the issue, and keeping them separate can help. If he returns to normal after being on his own, getting a larger cage to avoid territorial issues, adding in extra perches/barriers to prevent chasing, and additional food/water dishes can help with aggression. If A Rod continues to act ill even on his own, a vet visit is best.

Regurgitation can be a sign of affection/wanting to mate, and some birds may display this behavior toward people they are bonded with if they do not have a mate. It is best to discourage the behavior by either stopping interactions when it happens, or distracting with a favorite toy or treat instead. Puzzle toys or things that can be chewed on/groomed may help him get out some boredom or affection for the object instead.

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.

It's possible one or both birds may be ill, especially if they are not acting normal and seem very lethargic. Keeping them warm and eating and drinking can help, but it is best to bring them to your vet so they can be examined and any more serious issues such as illness or injury ruled out. If one seems very ill or injured, separating the two birds into hospital cages can help until you can get into your vet.