Just put them both in /etc/resolv.conf, or better - fix them to know of each other's zones. You can also run local DNS server, which you can tell where to look for what, but this is not a workstation's function - fix/link your DNS servers/network. At least, you can use /etc/hosts as it was in the beginning...
2. that there be some failover for network connections.
By failover I mean being able to define one interface as "primary". The primary interface would set the default gateway and all that global-unique stuff (including resolv.conf, without feature 1.).
When that interface goes down, those global settings are changed to the ones provided by another active interface. If the primary interface goes up again, it restores the initial configuration.
Why not just use DHCP?
This would be very useful for cases such as a laptop with wired and wireless networking. The "wired" connection would be the primary interface. The "wireless" connection would take over if the "wired" one goes down (they may be different networks, e.g. we have a totally open and untrusted wireless lan and our linux users can't just unplug the cable and move around, they are forced to restart the interfaces).
You can add 2/more default gateways. In this case linux uses 'Dead Gateway Detection' - see http://www.ssi.bg/~ja/dgd-usage.txt for example. I've never used two network interfaces with DHCP, so I've never tried this trick with DHCP, but you can :) ...