Detail information on running scripts on the recovery screen I am looking for more detail information regarding the process mentioned here:  

1 https://forum.xda-developers.com/showthread.php?t=2058944  

2 https://forum.xda-developers.com/showthread.php?t=2307355   

(1) says that selecting "apply zip from sdcard" in the recovery screen will run the script and perform the mentioned commands. I suppose at this point there is no concept of  android OS's root user? Is this correct? Am I allowed full access to all the files and folders in the sdcard and phone memory at this point?  

The script in (1) also mentions BusyBox. I know there is an app called busybox which provides some tools, but since the android OS has not loaded up to this point, I guess the busybox in the script represents something else? Wikipedia says busybox is an old set of linux tools, so does this mean it is present by default in all android phone firmware?   

The zip file containing the script in (2) contains a file called "busybox" in the zip file root (no extension, binary file) and the one in (1) doesn't. What does this mean? (1) and (2) seem to be for different phone models, but shouldn't all phones have the run_program() and delete() commands mentioned in those scripts? Would these scripts work on a different phone model? How much chance is there of the phone getting bricked? (I have stock firmware + Odin for my test phone and I am ready to format and reinstall if needed, I just want to know if there is a possibility of bricking it so badly that even Odin wouldn't recognize it)  

Regardless of whether I am able to mount the sdcard/phone memory and run programs on the sdcard or not, I suppose the commands in the script such as ui_print() should work on any phone. Where can I find more commands that can be used in such a script file, and more details on how such scripts actually work? The answer can be limited to samsung phones if there are no general answers.
Yes, from custom recovery there is generally a high risk of deleting everything in theory. Your data security is completely dependent on the developer's coding ability and your trust in him. Luckily a full backup should be sufficient enough to restore everything in case of damage.  

A flashable zip is usually pretty generic. in your example it's only purpose is modifying device specific things, so the reason is not the script language itself but simply the content of which apps will be removed. The first link is for indian apps, the second link is for samsung apps. Samsung is a good example because it has a separate partition for its bloatware - you need a flashable zip which delete apps from /preload instead from /system.  

linux executables doesn't have file extensions like *.exe - busybox/toybox/toolbox are just such executables. busybox is a multicall binary wich emulates most of linux commands like (ls cat dd grep find ...) in a single file. it's main purpose was to safe disk space. if you see any busybox app in Playstore it is always just a install helper for this binary.
android default is toybox (same as busybox but under apache license)
http://landley.net/toybox/faq.html  

Edify is a very limited script language only used in flashable zips
https://source.android.com/devices/tech/ota/nonab/inside_packages