There's an option to unhide old nodes in Houdini by typing 'opunhide' in the textport and choosing the relevant nodes. However, this process of 'opunhide' has to be done for every new project that I start in Houdini. How do I ensure that once I've unhidden a node, it stays unhidden forever even when I start a new project?

try creating hscript file 456.cmd in your $HOME/houdniversion/scripts,
and add some stuff you want, say, opunhide Sop *

Thanks. It worked. I created a new 456.cmd file in C:\Users\myusername\Documents\houdini19.5\scripts