Routing – HP 1910 disable layer 3 mode

Your problem is, indeed, with routing. Things in VLAN 1 are using 5.1 as the gateway, right? 5.1 doesn't know about any other internal networks. (i.e. VLAN 10) Likewise, the hosts in VLAN 1 don't know about VLAN 10.

When clientA tries to ping serverA, the packets go to the ASA and die as it doesn't know where to send them. (it's default route would send them to the internet!) When serverA tries to ping clientA, the packets go to the switch and then to clientA, but then the replies, as before, go away.

For this reason, it's common to keep the "routing hardware" and "hosts" in different networks. VLAN 1 being just the ASA and Switch. And then VLANs 10 and 11 for the two internal networks. The switch would then be the gateway for VLAN 10&11, with it's default being the ASA. The ASA would have to have static routes added telling it about the internal networks -- routed to the switch. Without this, either the ASA has to "hairpin" traffic back to the switch to reach VLAN 10, or every host in VLAN 1 needs its own route to VLAN 10 (via the switch.)

On the switch, your IP for interface vlan 20 is wrong.

# interface vlan 20
# ip address 10.1.2.1 255.255.255.0
# no shutdown

Then make sure you have ip routing on.

Also the gateway for PC A should be 10.1.1.1 while the gateway for PC B should be 10.1.2.1. The subnet mask should be 255.255.255.0 not the gateway.

Once fixed, make sure you can ping the gateway of each PC from the respective PCs as well.