How can I verify that the Bitcoin client is well connected to the network and is able to receive incoming connections?
Get your ip from http://whatismyipaddress.com/ and put it in here http://www.subnetonline.com/pages/network-tools/online-port-scanner.php with port 8333. If you can accept incoming connections you should see a result similar to below:
There is a port checker tool that is not browser based that will allow you to to check to see if the port 8333 is open. I had a situation with my windows 2008 R2 VPS where i was stuck at 8 connections. I went into the bitcoin client and unchecked "map using UPnP", shut down the bitcoin client, then went into windows firewall and added bitcoin to accept incoming TCP, UDP connections for the bitcoin-qt.exe. Then I restarted the bitcoin client, and I had 20+ connections after 10 minutes or so.
I didn't even have to use the port checker tool.