Okay, I'm using slackware.

Can someone please explain the relationship between ownership of a user's home directory and the user's ability to execute a program (e.g., startx)? I don't quite understand what a user's home directory has to do with program execution of a file that's not in that directory. For example, why can't I execute startx as "userx" unless "userx" is the owner of /home/userx? What does the user's home directory have to do with the ability of that user to execute a file? Or, mechanically, how are the settings of the user's home directory related to what that user can do? How are they linked? Does it have to do with the hidden files in that user's home directory?

startx initializes a process that need a write access to user's home directory. the program is bound to user's home directory and that's the way it's meant to be. There are no othere possible standard permanent places with writing permission on linux, so programs have to use user's ~ to write configs/other data, which is not meant to be temporary(that may go to tmpfs or /tmp) into them.