What exactly happens when questions/answers/comments are flagged?
Are they deleted after a certain number of flags? Or do they just become invisible? Are moderators notified? Is the author notified? Are they still listed in pages/searches?
When a post is flagged (question, answer, or comment), it goes into the flag queue, where the site moderators will review it. The post will be shown with all flags raised on it (including the type of flag, and any custom message with the flag).
A subset of flags (most of them) are also visible in the queue to non-moderator users with enough reputation.
Several actions can be taken when reviewing a flag:
The post owner is not notified, and the post will continue to appear on the site as normal. The only exception is that a comment will be automatically deleted if it is flagged enough times.
As far as deletion vs. invisibility: all deletions on the network are "soft" deletions. The content is hidden, and only visible to moderators and high-rep users. Only in rare cases is content actually deleted; this can only be done by Stack Exchange developers.
Author is never notified as per my experience.
But yes question is closed when any moderator closes that question or it gets certain closed votes.