Configure a DNS server per nic interface (eth0 / eth1)?

You can't easily do what you want.

Or how do I configure a different DNS Name Server for eth0 vs eth1?

The name lookup for a hostname happens through standard system libraries and isn't associated in any way with a particular "connection". In fact, at the time the DNS query happens, there is no connection, because your application hasn't even figured out the address to which it's going to connect (which is why it's using DNS in the first place).

How do I get it to respect the DNS settings in ifcfg rather than a default for resolv.conf?

The Linux resolver only has a single, global configuration (/etc/resolv.conf). There is no per-interface, per-domain, or per-connection setting of any sort. The settings in /etc/sysconfig/network-scripts/... are only used to populate /etc/resolv.conf, and generally if you specify DNS1 and DNS2 in these files, the last interface to come up will be what you see in /etc/resolv.conf.

Is there a better way of handling this?

Can you tell us what you're actually trying to accomplish? We might be able to suggest better solutions if you can tell us more about your specific situation.

A DNS request is basically either

1. "what's the IP address of host1.domain1.com," or
2. "what's the hostname of 192.168.0.5."

So there's no knowing at "request" time which Ethernet card is going to be involved. What you could reasonably ask would be "all requests ending in 'domain1.com' should go to 192.168.0.2, and all other requests should go to 10.0.0.2."

And likewise, "All reverse DNS requests matching 192.168.0.0/24 should go to 192.168.0.2, and the rest should go to 10.0.0.2."

As larsks said, Linux doesn't support such a configuration. However, you could run your own, minimal DNS server that implements the above logic, and forwards requests to the appropriate "real" DNS server.

I believe dnsmasq can do this (see How to configure dnsmasq to forward multiple DNS servers?). But before I found that out, I rolled my own in Twisted:

from twisted.internet import reactor
from twisted.names import dns
from twisted.names import client, server

class Resolver(client.Resolver):
  def queryUDP(self, queries, timeout=None):
    if len(queries) > 0 and (str(queries[0].name).endswith('.domain1.com'):
      self.servers = [('192.168.0.2', 53)]
    else:
      self.servers = [('10.0.0.2', 53)]
    return client.Resolver.queryUDP(self, queries, timeout)

resolver = Resolver(servers=[('10.0', 53)])
factory = server.DNSServerFactory(clients=[resolver])
protocol = dns.DNSDatagramProtocol(factory)

reactor.listenUDP(53, protocol, interface='127.0.0.1')
reactor.listenTCP(53, factory, interface='127.0.0.1')
reactor.run()