PANDAS (Pediatric Autoimmune Neuropsychiatric Disorders Associated with Streptococcal infections): Given the patient's history of PANS/PANDAS-like symptoms following a sinus infection, recent streptococcal infections (perianal strep and presumed strep throat), and the current presentation of OCD-like symptoms, rash, and vaginal itching after antibiotic treatment for a presumed streptococcal infection, PANDAS is a highly plausible diagnosis. The recurrence of neuropsychiatric symptoms following streptococcal infections is characteristic of PANDAS.

Streptococcal Intertrigo or Perineal Dermatitis: The rash on the neck and vaginal itching could be related to a streptococcal infection, especially given the patient's history of streptococcal infections. This condition can cause skin symptoms and could potentially trigger or exacerbate OCD-like behaviors in a susceptible individual.
Yeast Infection or Bacterial Vaginosis: These are common causes of vaginal itching and discharge in children. Although they might not directly cause OCD-like symptoms, the discomfort could potentially exacerbate anxiety or obsessive behaviors in a child prone to them.
Eczema/Dermatitis: The rash on the neck could be eczema or dermatitis, which can be itchy and uncomfortable, potentially contributing to the child's distress and exacerbating OCD-like behaviors.

Sexual Abuse: Although unlikely and not directly related to the medical history provided, any genital symptoms in a child must prompt consideration of sexual abuse. It's crucial to investigate and rule out this possibility due to its severe psychological and physical implications.
Neurological or Psychiatric Conditions: Conditions like Tourette's syndrome, other anxiety disorders, or even early signs of psychosis could present with OCD-like symptoms. These need to be considered and evaluated by a specialist.
Inflammatory or Autoimmune Conditions: Conditions such as lupus or rheumatoid arthritis can present with a wide range of symptoms including rashes and neuropsychiatric manifestations. Although rare in children, they are critical to diagnose early.

Autoimmune Encephalitis: A rare condition that can present with neuropsychiatric symptoms, including OCD-like behaviors, following an infection. It's a diagnosis that would require extensive investigation, including neurological evaluation and potentially brain imaging or lumbar puncture.
Other Infectious Etiologies: Less common infections that could cause similar symptoms, such as Lyme disease (if the child is in an endemic area) or mycoplasma infections, which can occasionally cause neuropsychiatric symptoms in addition to more typical respiratory or dermatological manifestations.