Hello !! I am dealing with the same problem as you.
The solution to the problem that half-life does not appear in the game selection, and only counter-strike, is as follows:
Go to the root directory of the game, that is, the folder where you have the half-life server installed and enter the folder of the mod valve. For example: "\ SteamCMD \ HLServer \ valve \" (enter the valve folder) and open the following file with notepad, "liblist.gam".
Edit it, and where it says: type "singleplayer_only", put: type "multiplayer_only".
This will show half-life when starting the Half Life Dedicated Server.

And as for the error window that says "error reading the application id, please either write it in steam_appid.txt or use the launcher!", It is because you have to put a file called "rev.ini", in the root folder of the game.

Here is a video that explains how to fix it: https://www.youtube.com/watch?v=HFknHY21rCc

PS: Sorry if you find misspelled words, I don't speak English, and I had to use the google translator, which is not perfect.