Hi Adam,
You can check Gilles' DC t-shooting guides that should give you a very good overwiew about packet processing on the ACE; also you can check
the Cisco wiki site where you find the scenarios plus a detailed explanation for traffic management.
Now going back to your issue, you problem can be splitted in two parts.
1. Web server not able to ping VLAN 101 ACE's SVI.
ACE is a closed device, meaning that access to each Interface/VLAN needs to be explicitly configured; you need to apply the management policy
to the 101 SVI to allow ICMP or any other management protocol. You can apply the same (service-policy input MGMT-ACCESS) or create a new
one just for ICMP, that's up to you.
2. Web servers not able to communicate with APP servers thorugh VIP.(vise-versa)
Problem here is that servers are trying to communicate through SVI 101 but no VIPs are applied to it so the ACE will simply discard the packets
for 10.10.12.20/10.10.12.21 on that interface, servers have the ARP and everything to reach those VIPs but the ACE has not been instructed to do
load balancing for clients reaching it out through VLAN 101.
In order to do load balancing between APP & Web Servers you need to configure L4-CONTEXT-A-VLAN on SVI 101 as well.
Also since your servers are sitting all in the same VLAN you're going to need client NAT to prevent assymetric routing on server-to-server communications.
I've attached a sample with NAT based on your config.
HTH
__ __
Pablo
