Appointing of auditor by directors

Details will depend on jurisdiction so you don't need to know them. For example, the Companies Act may specify that the general meeting be called within 3 months of the directors making an appointment. If an AGM is only 4 months away, that would suggest that the year end is imminent (if not already passed!) so the directors would call a general meeting ASAP to make the appointment in order that audited financial statements be presented at the AGM.

1 - short answer "yes". ANNUAL General Meeting is the compulsory meeting for important business like appointment/reappointment of auditors/directors and declaring the final dividend (if any) for the year (ended) based on the audited financial statements which are formally "laid before the members". GMs should be called by the directors as frequently as the circumstances of the business demands - attendance is not compulsory - resolutions will be passed (or not) based on those attending to vote. But again, the specific details of any jurisdiction are not examinable. Under the UK Companies Act - for example - there is a "written resolution regime" available to private companies (only) under which most resolutions can be passed without a physical meeting of members. So your point 2 is not an issue.

"general meeting" means a meeting of the shareholders. The AGM refers to the meeting that must be convened annually to pass resolutions on the types of things I have mentioned. Additional GMs USED to be called "extraordinary" GMs (in UK law) - but there's nothing extraordinary about them - they are just any GM that is not the AGM.