It would be OK to use HTTP status codes, but it depends on who is consuming your API. Sometimes it is better to just return 200 OK and then include Error information in the body.
With Status Codes
If you go with status codes just return the first error encountered, no use in handling the request further anyways, so in pseudo:
if (request is not POST) return 405; //abort here
//we know request is POST here
if (request not auhtorized) return 401; //abort here
//we know request is POST and authorized
if (request requests a not exisiting entity) return [404, 422, ..., 5xx] either will do; // abort here
// we now know the request is POST, autorized and requests valid information
processRequest();
Without Status Codes
As an alternative, since you tagged ajax, I assume you are returning JSON, so just return 200 OK and include a the fields success : [true|false] and errorMessage : ["Not POST"|"Bad Auth"|"Bad Request or Unknown resource"|"OK"] in your JSON answer.
You could also combine both ways, but depending on the ajax client not all will work well with all status codes. Given the information in the answer, all you need to do is check if success === true and handle error otherwise.