Regarding your second cluster of questions: Everything a computer does is ultimately done by hardware. But we can make some distinctions between different kinds of hardware within the computer, and different kinds of programs interacting with them in different ways.
A modern computer can be divided into three major components: central processing unit (CPU), main memory (random access memory, RAM), and "peripherals" such as disks, network transceivers, displays, graphics cards, keyboards, mice, etc. Things that the CPU and RAM can do by themselves without involving peripherals in any way are called computation. Any operation involving at least one peripheral is called input/output (I/O). All programs do some of both, but some programs mostly do computation, and other programs mostly do I/O.
A modern operating system is also divided into two major components: the kernel, which is a self-contained program that is given special abilities that no other program has (such as the ability to communicate with peripherals and the ability to control the allocation of RAM), and is responsible for supervising execution of all the other programs; and the user space, which is an unbounded set of programs that have no such special abilities.
User space programs can do computation by themselves, but to do I/O they must, essentially, ask the kernel to do it for them. A system call is a request from a user-space program to the kernel. Many system calls are requests to perform I/O, but the kernel can also be requested to do other things, such as provide general information, set up communication channels among programs, allocate RAM, etc. A library is a component of a user space program. It is, essentially, a collection of "functions" that someone else has written for you, that you can use as if you wrote them yourself. There are many libraries, and most of them don't do anything out of the ordinary. For instance, zlib is a library that (provides functions that) compress and uncompress data.
However, "the C library" is special because (on all modern operating systems, except Windows) it is responsible for interacting directly with the kernel. Nearly all programs, and nearly all libraries, will not make system calls themselves; they will instead ask the C library to do it for them. Because of this, it can often be confusing trying to tell whether a particular C library function does all of its work itself or whether it "wraps" one or more system calls. The pthreads functions in particular tend to be a complicated tangle of both. If you want to learn how operating systems are put together at their lower levels, I recommend you start with something simpler, such as how the stdio.h "buffered I/O" routines (fopen, fclose, fread, fwrite, puts, etc) ultimately call the unistd.h/fcntl.h "raw I/O" routines (open, close, read, write, etc) and how the latter set of functions are just wrappers around system calls.
(Assigning the task of wrapping system calls to the runtime library for the C programming language is a historical accident and we probably wouldn't do it that way again if we were going to start over from scratch.)
