Let me try to help you in this special network design. You don't say why you can use a router and what you want to achieve, so I start unraveling your design.
The first thing to mention is the difference between a public and a private address. If you want to communicate between those private and public addresses, you need a router.
I am attempting to use a Galaxy Tab A in a local private network with no router
If you don't have a router device, you need a other routing functionality.
There is something like a routing table. Pointing to the address 0.0.0.0 in a device means that you want to reroute all the traffic which is not on your local network. The routing table address 0.0.0.0 normally points to an address which is capable to reroute the local traffic on the private network to the public network as a "routing" device or shortly "router", which you say that you don't have.
I am able to ping 192.168.12.9 and 192.168.12.10 from my win 7 pc.
A successful ping is only possible if a device is on the same private network or if there is a router connected to that network (which your claim is "there isn't). Now if, if it is possible to use a Win 7 PC to ping your devices connected on the private network and the same Win 7 PC can connect to the public Internet at the same time and network, there must be some "router" somewhere in your tested network.
The simple solution is to open a CMD box (Open start and type cmd) in the Win 7 PC. In the CMD MSDOS box type ipconfig a look to the active network interface. There you found the default gateway address which point to the router device function.
A point to the DNS address, this should be point to a DNS server in your private or public network like Gooogles DNS server on 8.8.8.8. In a private network with a router you will see that some configurations points to the default gateway address in those cases that the router reroute the DNS address to a DNS server in the private or public network.
If this explanation is not the solution, we can only help you if you explain more precisely what you want to achieve.