C isn't a program, or something you install; it is a programming language for writing software. Linux and most of the programs that run on it were written in C, and a C compiler and libraries are supplied with most systems, or are readily available.
Linux supports any written language: it understands Unicode natively, so it can display the characters of any language with the appropriate locales included. As far as programming languages, Linux is written in C, but almost any language, from assembly to C to C++ to Python to Perl to .NET can be used on it.
c
Very little. C is a very common language to write operating systems in.
Most programs in Linux are written using C or C++.
C is a programming language. A shell is an interface.
It's 1405 in my Linux box.
There is very little difference in the C compiler between Unix and Linux; in some cases (the gcc compiler) it is the same. The differences come in when using system calls; some system calls do not exist in Unix or Linux, although most do. The program I work on compiles the same way (for the most part) between all commercial versions of Unix and several variants of Linux. In other words, the code is fairly portable across platforms.
Linus Torvalds wrote the Linux kernel in C. Other components written by other contributors for other distributions may be in C or C++.
For a start, you can install a C-compiler.
dnf install ncurses-devel
Yes, if you install 'gcc', 'make', 'glibc-dev' and a few similar components (packages) -- most likely the are already installed, or can be installed easily.
There are many operating systems developed in C, the most notable of which is Linux.