How do I install and use the Half life 2005 package on Arch Linux? I can't run batch files. I tried adding the hl.exe as a non-steam game in steam but that didn't work. I tried adding the first (Marked with '1.') .bat file that the instructions told me to run as a non-steam game but that didn't work either.

Half life anniversary edition works without issue through steam. Older versions also work fine through the official steam game in my library.

Make a file called steam_appid.txt in the same directory the bat files came in (you shouldn't mess with file placement at all in the package besides creating new files) and add just the number 70 to the newly created file, that should work

After creating the steam_appid.txt file, should I run it as a non-steam game? Or should I try to run the batch file directly with wine or something similar?

run it as a non steam game with enabled compatibility with proton, it should work with recent versions of proton I believe