Central Kurdish

No — they're separate Kurdish languages, related but not mutually intelligible. Sorani (Central Kurdish) is spoken in Iraqi Kurdistan and western Iran. Kurmanji (Northern Kurdish) is spoken in Turkey, Syria, and parts of Iran and Iraq. Sorani has dropped grammatical gender and uses ergative constructions in the past; Kurmanji preserves a two-gender system. Speakers of one need real study to follow the other.

A modified Perso-Arabic abjad written right-to-left, extended with letters to mark vowels that the basic Arabic script doesn't represent (since Arabic is consonant-focused with optional short-vowel diacritics, Sorani Kurdish needs explicit vowel letters). The script is mostly phonemic — what you see is what you say — once you learn the extended letters.

In past tenses with transitive verbs, Sorani treats the agent and patient differently from how it treats them in present-tense or intransitive constructions. The verb takes agreement clitics that mark the agent in past transitive sentences. The pattern is reminiscent of split-ergative systems in Hindi, Pashto, and Marathi, and reflects Iranian languages' broader tendency toward ergative-style past tenses.

Iraqi Kurdistan (the autonomous Kurdistan Region in northern Iraq), where Sorani is the standardized administrative language used in Erbil, Sulaymaniyah, and Halabja. In Iran, Sorani is spoken across Kurdistan and Kermanshah provinces. Diaspora populations exist in Turkey, Syria, Sweden, Germany, and the United Kingdom.

Yes — both Iranian languages, though in different subgroups. Sorani belongs to Northwestern Iranian; Persian is Southwestern Iranian. They share grammatical structure (mostly SOV, gender-free in modern forms, similar tense systems) and many cognates, but the languages aren't mutually intelligible. Kurdish has retained features of older Iranian that Persian has lost, including the ergative-style past.