Haproxy not logging with rsyslog
The rsyslog configuration assumes a chroot'd HAProxy, which does not match the haproxy config. Either chroot HAProxy by adding the line
chroot /var/lib/haproxy
to the global stanza of the haproxy config, or change the location of the socket rsyslog creates
$AddUnixListenSocket /dev/log
Under global change/add:
global
chroot /var/lib/haproxy
log /var/lib/haproxy/dev/log local0
log /var/lib/haproxy/dev/log local1 notice
... rest of your file