APM site slowness/timeout

Client request traffic should arrive at the external VLAN of the LTM (where the LTM VIP lives). If you run a tcpdump on the LTM and filter for traffic to the destination address (the VIP address) you can see whether or not client requests are making it to the VIP when there is a failure. You're specifically looking for packets arriving at the VIP address when it's failing. If you do see packets arriving to the VIP, then you can mostly rule out any network issues and focus on the LTM/APM configuration.