Right, howver the centos haproxy needs to be able to lookup the actual IP address of the Google server, otherwise it will point itself and it will never work.
So make a ping from the haproxy centos machine to connectivitycheck.gstatic.com and make sure it goes to Google, not itself.
Yes.
No.
Use haproxy logs:
Use curl -vv both on haproxy centos box (not redirected) and other machines (redirected), check what happens in both situations and also check haproxy logs.
Keep it.