Pass the player name on the command line when launching the game

The best option might be Condition Zero instead of plain CS 1.6

Z-Bot works still fine with the Steam Version:

• Download and extract the ZBot ZIP-File
• Copy all files (except liblist.gam and commandmenu.txt) to your ...\Steam\SteamApps\\..\cstrike directory
• In the Steam Game Library -> right click on Counter-Strike -> Properties -> Launch Options and add -dll "dlls\zbotcz"
• Now you can run the game

I used to get these problems in my college when arranging CS 1.6 events. However, we got over these by making other PCs server or re-installing the game. This can happen due to a corrupted file too. Try re-installing the game and checking your graphics drivers and also the Hub/Switch which you use, make sure enough bandwidth is allocated in the configuration. Mostly this happens either due to a bad connection medium or hardware.