Are fine particles of dust and dirt that may enter the mouth and are difficult to remove considered unavoidable, and is it permissible to swallow them intentionally, knowing of their presence, without invalidating the fast?

Swallowing particles that are difficult to avoid—such as the dust from prostration—does not affect the validity of the fast. It is feared that your sensation of them might be a whisper (waswasah). Your fast is valid, and you must beware of giving in to obsessive whispers. If you find something of that in your mouth, it is not permissible for you to intentionally swallow it.