You are natting that traffic at the router to the routers outside interface if you check the logs on the ASA when running that ping you will see nothing coming from the 192.168.1.0 network
He won't be able; to the only IP address that's on the ASA is the 192.168.0.0/24 network and whatever DHCP he's getting from the ISP. The reason he's able to ping one way but not the other is that he has NAT configured on the router so anything sent ...
You have NAT configured and the ASA so all the traffic from the LAN looks like it's coming from the routers outside IP address (which is in the 192.168.0.0/24 network) unconfigure that NATs on the router put a static IP address (such as 192.168.0.2) ...
Based on that diagram if 192.168.0.3 can talk to 192.168.0.12 by being on the same vlan there is no reason to NAT traffic. Now as to why 192.168.10.2 is working but 192.168.10.3 is not is because on the router I think proxy arp maybe enabled so the r...