Getting a better handle on APC Cache Bins / code for settings.php
"APC has been running for 3 min 54 sec. Currently caching 0 entries (0
  bytes)."


This message suggests that your APC is running for each php request separately ultimately not caching anything at all.

I had a similar issue and it required to set up PHP once again but in a different mode.

APC works fine for multiple processes with apache/2 + php 5.3.x + php-fpm + fcgi in FPM/FastCGI mode.

PHP 5.3.x is configured with following options --enable-force-cgi-redirect --enable-fastcgi --enable-fpm

In APC section of my php info I can see:

APC Support = enabled
Serialization Support = php
MMAP File Mask = /tmp/apc.something  | where something is a random 6-character string

Please check your phpinfo 

In PHP Configure Command you should have:
'--enable-fastcgi'
and
'--enable-fpm'

In Server API :
 FPM/FastCGI

Check if you have cgi-fcgi section and if you can find 
php-fpm set to "active"

NOTE: There might be another solution if you use php 5.2.x or older.