I'm sorry that your question wasn't answered in a timely manner. We don't have many avian vets on this site. Adusting how you pick Blackie up should avoid her emptying her crop as she's been doing. I suspect that you're compressing her crop - the squishy dilatation of her esophagus - when you're lifting her. If Blackie is vomiting "unaided", however, she's a sick chicken and needs the attention of an avian vet (please see here: www.aav.org).
Her gaping ("beak open..breathing out her mouth") is particularly concerning because it represents respiratory compromise. Her vet needs to carefully examine inside Blackie's oral cavity for inflammation and growths and also consider respiratory infections involving Blackie's sinuses, air sacs and lungs. The gapeworm is also a consideration but much more commonly seen in waterfowl.
Unfortunately, the symptoms you've mentioned can indicate any number of illnesses or health issues. In avian medicine, there's rarely one cause of a condition, so we usually begin with a list of differential diagnoses and use lab tests, X-rays, and physical exams to differentiate one from another. With this in mind, your best course of action is to reach out to your county-extension poultry personnel or avian-oriented veterinarian for help in differentiating the various causes of what you're seeing. Veterinarians can perform a physical exam and run diagnostic tests, including X-rays, to distinguish between the various etiologies.
It's best to approach the diagnostic process with a clear sense of Blackie's financial value to your operation. Although some services such as your county animal disease diagnostic laboratory might be available free of charge through a county agency or land-grant extension office, the expense of some diagnostic tests and treatments can add up quickly. While it’s always worth your time and money to identify a bacterial or viral infection that could potentially impact more than one member of the flock, this might not be the case with a condition that only affects one hen.
Please respond with further questions or concerns if you wish.