I know this thread is old, but it is still one of the top links on Google when running a search for using GCP for Half-Life as of today. I came here looking for a solution to this problem, and since I could not find one, I began experimenting on my own.
My problem was with Half-Life 1, and I did find a full solution. Here is how I got it to work on HL1:
From the main game screen (or from the pause screen), choose Options, then the Mouse tab, and make sure the "Raw mouse input" is selected.
I had a similar problem to the OP with the right stick causing me to look down and then spin in circles. I was able to fix this by toggling "Steam Mode" off in the GCP settings. However, when I made this change, rather than making me look at the ground, whenever I would try to turn in any direction (especially if I turned quickly) it would turn turn more than 90 degrees it would snap me back to the direction I had been looking when the game loaded, essentially preventing me from actually turning in a meaningful way. The "Raw mouse input" change that I made fixed this issue, regardless of whether "Steam mode" was active or not.
I did not have this problem with HL2, with or without "Steam mode" activated. I am using GCP 3.3.1 at the time of this post.