DNS does not resolve after VPN

When you are connected to your VPN, DNS queries are being handled by your office's servers. When you query for "router", your computer is adding the suffix pacs.local. For whatever reason, your office's DNS servers appear to respond with 66.114.124.140 for any domain they don't know about.

Try running nslookup router 10.3.1.120, you should get back the correct IP address for your router because your local DNS server will be doing the address resolution.

It sounds like what you want is the same as the asker in this Server Fault question: you have multiple NICs on a computer (one is a virtual NIC), and you want DNS resolution to first try your local DNS servers, then try the remote DNS servers if that fails. That person's issue was resolved by changing the order of network adapters as described in Microsoft KB894564, so perhaps that will work for you as well.

On the IP level, What you are looking for is called Split Tunneling, and it is configured from the client (at least in non-managed scenarios). it allows requests to local lan addresses to use your local (logical) network, and requests to remote lan addresses pass through the VPN tunnel. if split tunneling is not enabled, the host connected to the VPN will not be able to communicate with other machines in the local lans IP range at all, so its more than just a dns resolution issue. with split tunneling enabled, if you will be accessing the Intranet, your default route must point to a gateway on the local lan, or there will be ambiguity as to which network to use.

the issue here however is DNS. windows (and linux as well) tie the DNS server configuration to the local interface, not to the individual routable connections, meaning that you will only ever be able to use one DNS server at a time and cannot select one conditionally based on destination address. Try looking at running DNSMasq on the VPN'ing PC as described here: http://blog.nixpanic.net/2013/03/use-dnsmasq-for-separating-dns-queries.html