This is what it looks like now
The symptoms you've described, such as scabbing, hair loss, sticky skin, redness, and excessive itching, along with vomiting, suggest that there might be an underlying issue that needs to be addressed. Please understand that it is difficult to determine his exact condition and cause from the photo.
The skin problems could be due to several factors, including allergies, infections (whether bacterial, fungal, or parasitic), or even an underlying systemic condition. Bulldogs, in particular, are prone to skin issues because their skin folds can harbor bacteria and yeast.
Firstly, to prevent further irritation, consider using an Elizabethan collar (cone) to stop your dog from scratching and licking the affected areas, which can worsen the condition.
Secondly, bathing your dog with a medicated shampoo containing chlorhexidine or miconazole can help reduce bacterial and fungal infections. Make sure to leave the shampoo on for 5-10 minutes before rinsing thoroughly.
Additionally, after bathing, you can apply a thin layer of antibiotic ointment, such as mupirocin, to the affected areas.
If his symptoms persist, or if you notice diarrhea, loss of appetite, seizures, or metastases to other parts of the body, I recommend taking him to a veterinarian to determine the exact cause and treat accordingly.
If you have any further questions, please feel free to ask. Thank you.