How do I test that blockchain p2p replication is properly, securely exposed to the internet? I have bandwidth and CPU to spare and I'd like to participate in the P2P nature of sharing the blockchain to other peers.  How can I properly, and securely participate?

Also, how can I test or verify my client that its working correctly?

Can I participate in the same (port 8333) blockchain replication if I run in server mode?
If you already have bitcoin up and running for some time, just open terminal/commandline window and run:

netstat -b (Windows)

netstat -tnp (Linux)

In the Local address column output, note whether there is port 8333 used with any bitcoind connection. If yes, you know that your node is reachable from internet and thus you are fully participating. Otherwise you need to allow connections to port 8333 from outside in your firewall, so that you don't have to depend on other nodes accepting connections. Also, healthy and up to date node is sending/receiving transactions quickly.

Other concerns, like security, are handled by bitcoind automatically, as it verifies all received data. The bitcoin-qt GUI is working in exactly the same way, I just presume you'd prefer running bitcoind as a background service.