Hi Eric,
HOST_C and the NIC facing it on HOST_B should be on one subnet, such as
10.0.0.3/8 for HOST_C and 10.0.0.1/8 for HOST_B.  HOST_A and the NIC
facing it on HOST_B should be on a different subnet.  You're setting up
HOST_B as a router and so each side of the router will need to be a
different subnet so it can make decisions about where to forward packets.
In RedHat 7.1, make sure you have the proper network information saved in
/etc/sysconfig/network and /etc/sysconfig/network-scripts/ifcfg-eth0 and
/etc/sysconfig/network-scripts/ifcfg-eth1.  You said you enabled packet
forwarding, so that's good.  Next, you'll need to setup routing rules for
each subnet using the route command.  Type "route" by itself to see what
is currently being done.  In the Destination column you have the network
that you are trying to reach.  In the Gateway column, you have the IP
address of the next hop to the network in the Destination column.  So, to
add a route for a 172.16.0.0 network with a 24 bit netmask (255.255.255.0)
that uses 172.16.0.1 as the gateway, you would do this:

route add -net 172.16.0.0/24 gw 172.16.0.1

You can then save these routes to:

/etc/sysconfig/static-routes

The entries in static-routes are passed to the route command on startup in
this format:

ethx net ip-address netmask 255.netmask gw gateway-ipaddr


By the way, the struggling that you're going through right now is a good
thing, trust me.  You're on the verge of really understanding networking
and routing and it's all going to gel for you soon.  Keep at it and this
will all soon be a piece of cake for you.
~M