There have been two versions of SWIS in Orion for the past couple of years: v2 and v3. You almost certainly want to be using v3 for everything at this point. The HTTPS URL for SWISv2 is this:
https://servername:17778/SolarWinds/InformationService/OrionBasic (notice no "v2")
The HTTPS URL for SWISv3 is this:
SWQL Studio can be used to connect to either version. In addition to the HTTPS/SOAP protocol used in the CSClient sample, SWIS exposes the same API through a Microsoft-proprietary protocol called net.tcp (somewhat better performance than HTTPS/SOAP, but only works with .NET applications). SWQL Studio uses the net.tcp protocol by default.
SWISv3 also supports the same API using a REST/JSON protocol at https://servername:17778/SolarWinds/InformationService/v3/Json. See SWIS REST/JSON API for details.
First I want to thank tdanner for all his time.
I hope what I found will be helpful to someone, but it may be a little unique.
As best as I am able to confirm there was a combination of our network setting, a self-signed SSL certification, and interaction with the vhost configuration file that caused the problems I encountered. The most important being (I think) the network settings. When I was finally able to get to the TCP IP error 10060 I concluded (not absolute) the network was stopping the HTTPS connection set up on by dropping a response packet. I moved the built app to another part of the network, and after a little work was able to connect to the server and download and xml response.
Thanks again tdanner, your help enabled me to rule out certain issue while debugging.