Are you trying to ping at the IP level, or the Ethernet level? Some devices (for various reasons) aren't going to respond to a ICMP ping.

Is there an easy way (like via a .Net library) that allows me to get the IP address and subnet mask of the local machine that the application is running on?

A subnet allows the flow of network traffic between hosts to be segregated based on a network configuration. By organizing hosts into logical groups, subnetting can improve network security and ...