Openvpn – Cisco Anyconnect via OpenVPN tunnel on Windows

As mentioned in a comment, figure out your split-tunneling (routing) first. The endpoint (Concentrator, ASA, PIX, etc.) admin usually has full control of how the client will handle this; some companies will only tunnel their network (so your client's DNS would go over the ISP) and some require tunneling all traffic over the link (so DNS would go over the link to corporate). It's a business/IT decision most of the time.

Peak at the routing table on the client and see what the gateways and such look like and where the traffic is being routed in their specific instance to provide the beginning point for your debugging. You might then try direct DNS queries from the client against a public resource (i.e. Google's new DNS thing) to see what the results are, while using TCPview (if your client is Windows) type software to watch the bits fly around.

You don't appear to have a route back to the remote host accessing the VPN. You need to add a static route for all traffic back to the providers gateway address:

ip route 0.0.0.0 0.0.0.0 <your default internet gateway>

That would cause the "No valid adjacency" message in the log. Hope that helps, let me know how you get on.