SharePoint Authentication from C# Application

I have this working now, and the issue seems to have been related to the URL used when you get your client context object. Our SharePoint implementation has multiple levels of sites within it. We have separate sites for every customer, and then a top level site. I was attempting to get a file from a library under a customer site, and I had been using a URL that corresponded to the customer site. Once I changed the code to use the top level site (literally just the domain name) for the client context, the popup browser started getting the FedAuth cookie and things started working. So either I don't properly understand how our SharePoint is implemented, or the authentication method is extremely sensitive to the URL you use to get the cookies. The absolute URL to the file I was attempting to copy down looks something like this:

https://mycompany.sharepoint.com/customers/theclientname/797%20Project%20Name/In%20Process%20Templates/standard.xml

This URL is a valid site:

https://mycompany.sharepoint.com/customers/theclientname

But when I used that URL with AuthenticationManager.GetWebLoginClientContext(), it failed. Switching it to use:

https://mycompany.sharepoint.com

to get the client context fixed the problem. Now, if permissions were different at these different levels, I suspect this would have failed - it would have gotten a valid client context, but suspect I could still have gotten a permissions error when trying to use OpenBinaryStream() on my file object.

This was a frustrating journey, and I was disappointed to not get any feedback from anyone else. Apparently my limited understanding of SharePoint and how to integrate to it is not as common as I thought it might be.