A sick leave application should state the dates and that you are unwell — you do not have to disclose your diagnosis. Mention whether you can supply a medical certificate, since most policies require one beyond two or three consecutive days.
Planned leave usually needs one to four weeks' notice depending on company policy and how long you will be away. Sick and emergency leave is notified as soon as you reasonably can.
For annual leave, generally no. For sick, emergency or statutory leave you may need to state a category — but not private medical detail.
Your line manager, with HR copied. HR records the entitlement; your manager approves the cover.