Apple - Terminal Closes Immediately

Your default shell is the brew-installed zsh. You probably removed the folder containing zsh and some other important files/folders/libraries.

In the preferences of Terminal (or System Preferences > Users & Groups > ctrl-click user_name > Advanced Options... > login shell | as far as I remember you have to reboot to make this work) set the default shell back to /bin/bash. Then relaunch Terminal and uninstall brew and reinstall it afterwards. You may have to uninstall brew manually by deleting some folders in /usr/local/ if you deleted too many files and folders previously ("I accidentally deleted some files from my Applications directory. I don't recall exactly what but some of them included the Cellar directory.").

In a second step reinstall brew and brew installed execs.