In Ubuntu you will find Coreutils installed .
This package contains the basic file, shell and text manipulation
utilities which are expected to exist on every operating system.
Specifically, this package includes: arch base64 basename cat chcon
chgrp chmod chown chroot cksum comm cp csplit cut date dd df dir
dircolors dirname du echo env expand expr factor false flock fmt fold
groups head hostid id install join link ln logname ls md5sum mkdir
mkfifo mknod mktemp mv nice nl nohup nproc od paste pathchk pinky pr
printenv printf ptx pwd readlink rm rmdir runcon sha*sum seq shred
sleep sort split stat stty sum sync tac tail tee test timeout touch tr
true truncate tsort tty uname unexpand uniq unlink users vdir wc who
Core GNU utilities .
You will find most of the basic commands from Core Utilities description page, said that , Linux is just a Name of Kernel, Whereas Ubuntu is an Entire Advanced Integrated OS of such utilities. As in Question
Such as cp, ls, grep, ps...
Are they stand-alone applications pre-installed on the operating
system or the innate part of the operating system?
They together actually forms Ubuntu/Linux Operating System , Either of them existing standalone doesn't help much.
Shell is basically Bridge between kernel and the user, a Command Interpreter where user can type command and the command is conveyed to the kernel and it will be executed.And it is Different from Kernel , refer basic image below
And for more descriptive image
File Manager mostly forms the Front-End of common Core Utilities.
So how does that
innate thing happens , its because ,
Linux is Monolithic kernel where all services (file system, VFS, device drivers, etc) as well as core functionality (scheduling, memory allocation, etc) are a tight knit group sharing the same space. It provides the shortest path to a working, stable kernel .Whereas
Memory, Process and Device managing are the functions of any Kernel.
There is also a very good description provided here at How to understand the Ubuntu file system layout?
Additions related to Libraries-
The GNU C Library and GCC GNU Compiler Collection is most important Library and Core requirement of Ubuntu/Linux Operating system. One of the reasons is Kernel is written Primary in C and hardware specific code using Assembly language.
To some extent
"OS is made up of two parts, i.e., Kernel and Shell" holds true , for example Servers , and Minimal Linux based Distros. But for a fully fledged Ubuntu Desktop you need more then just Kernel and Shell.
Gtk (Gnome) is based on C and C++ , Qt(Kde) uses mostly C++ i guess , they don't have any libraries in common. Which is why they provide different User experience.
For utilities ( packages ) , you will find Python pre-installed in Ubuntu & Ubuntu-distros , to provide extended functionalities to python based Applications ( Like Window managers i guess) and Scripts . As i mentioned early
"They together actually forms Ubuntu/Linux Operating System , Either of them existing standalone doesn't help much."
You can further refer the question How can I know if my an application on my desktop environment is loading other DE's modules, libraries, etc? .