The CD version of the Half-Life(The Firste) was a standalone game (pre-steam). It has absolutely no idea about steam. Launch by double-clicking the exe file. It will ask for the CD-Key found on the CD-case insert and play. Then install the mod to that copy of Half-Life.
By the way, next time you start steam it may detect the legit half-life and add it to steam forever. Then you can install THAT version too if you want provided no other steam account "saw" that installation/CD-key.

I deleted everything, reinstalled Half Life to its default location (C:\SIERRA\Half-Life\) ran it to make sure it worked, then installed the Cry of Fear mod to the above directory. When I double clicked the launcher shortcut on the desktop and hit launch i get an error window popping up saying "The system cannot find the file specified" along with some lines above it saying that if i hit continue it will ignore it an dtry to run or hit quit to cancel everything.

When I run HL.exe and click the custom button, select Cryoffear and hit activate and start a new game, another error window pops up with "Could not load library C:\SIERRA\Half-Life\cl_dlls/client.dll

mods got their own folders. I think you installed the mod on top of half-life instead of in a subfolder.
For example TFC goes in Half-Life\TFC\
Counter-Strike's cd install put it in its own folder, but if you move cstrike into the half-life folder it shows up in the activate/deactivate screen of half-life.
I think you put the contents of COF into the Half-Life folder otherwise it should work.
You'll have to fix half-life so it can run again so another clean install may be required.

Of course COF may only work on the original half-life that comes from steam...