Counterstrike will be too intensive on the HTML5 Servers. Here are the steps I took to get Dota 2, a different steam game, to work portably. Also, at the time of your post, it appears the Video Card Cameyo Servers are using does not support Directx, which make sense since they aren't meant to substitute your gaming pc.
1. Use a Virtual Machine to install the game on a completely clean system using Snapshots. If possible, package it on the oldest os you plan to use it on. (Meaning if you plan to play the Virtual App on Xp, capturing on Windows 7 is not recommended and can cause issues with dependencies.)
2. Once you have installed the game, configure it with whatever default settings you wish. Play at least one full game before doing the Post-Install Snapshot. This should ensure all dependencies are properly packaged.
3. Take the Post-Install Snapshot. Wait for it to finish. Then open it in package editor. Change the File System and Registry to Strictly Isolated and save. It should work now.
One thing to note, when I did this for Dota 2, I had to have a fast flash drive or the FPS would suffer severely because it couldn't load the files it needs to within a reasonable amount of time. CS 1.6 is a much older game so you should be fine, but I wanted to make sure you were aware.
Have you tried setting the entire file system for the virtual app to Strictly Isolated. This will make sure all changes are captured by cameyo. This should fix the issue with your masterserver and others.