Reseller hosting is, essentially, wholesale hosting. You buy a large block of hosting resourcing at below-retail pricing, and sell it at a markup to regular hosting customers.

The easiest way is to get started is to search for reseller providers and find one that offers the type of hosting plans you would like to sell.

Most hosting companies that offer reseller hosting also provide tools for setting up your online business, such as AccountLab Plus, ClientExec, and WHMCS.

A web server is a software application that communicates between your computer or server and the internet at large.
Web servers are a critical piece of a website hosting platform — without a web server, there is no website at all.
The web server runs continuously on the computer (server) that hosts your website. It listens for requests coming from the internet, and then responds to those requests by delivering web pages, images, scripts, and other resources back to the requester.
For static pages and resources, the web server simply finds the file and sends it on to the user who wants to look at the web page or view the image. With dynamic web applications and sites like WordPress, the web server works together with a language interpreter to read PHP (or Ruby, Python, or another language), run the application, and then return the output back to the user who is trying to read your blog or buy from your ecommerce store.

Most often, Apache.
Apache is the most common web server. It is free and Open Source. The vast majority of commonly-used web content management systems (like WordPress, Drupal, and Joomla) and other similar applications are written primarily with Apache in mind.
Within a Linux environment, there are two major competitors to Apache, but they each have their own cost associated with them:
LiteSpeed is the easiest replacement for Apache. It provides faster performance and better scaling, and it acts as a "drop in" replacement — which means that it (essentially) works the same as Apache. Anything that is compatible with Apache will be compatible with LiteSpeed, and the configuration settings should be mostly identical. However, LiteSpeed is proprietary software, so you have to pay for it.
The other popular alternative is Nginx. Nginx is faster than Apache, especially when serving static files or cached files, and can be made to be as fast (or faster) than Apache for interpreting PHP and other languages. Nginx is also free and Open Source, which makes it comparable to Apache from a financial standpoint. However, Nginx doesn't work the same way the Apache does. There is no guarantee of compatibility. Even when things are compatible, it can be difficult to find information — for example, it is possible to run WordPress with Nginx, but all the official tutorials and support documents assume you'll be using Apache.
Apache also runs on Windows, but most people running Windows server will use IIS, which is Microsoft's proprietary web server.
Some web applications and development frameworks, such as Ruby on Rails, have a built-in web server, and don't require a separate web server.

Apache is a web server.
Apache is the most common web server in current use. It open source and freely available to use without cost.
Apache forms the "A" in the somewhat famous LAMP stack: Linux, Apache, MySQL, PHP. This is the software platform most commonly used for many popular web applications, such as WordPress, Drupal, and Joomla.

Probably. Apache web server is the most popular one used in the world.
If you have purchased a shared hosting plan, or a preconfigured VPS, you almost certainly have Apache installed already.
If you have a dedicated server, or certain types of minimally configured cloud hosting, you may need to install Apache yourself.
Usually (but not always), if a hosting provider offers something other than Apache (such as Nginx), they will make it a major selling point in their advertising, and you would have noticed.

No.
You need a web server of some kind, but it does not have to be Apache. You could use Nginx, IIS, LiteSpeed, or another one.
Some development frameworks (such as Ruby on Rails include a built-in web server, so you don't need a standalone one.

Usually, but not always.
If you are trying to set up a simple WordPress blog, or another similar PHP-based app, and you have only moderate (or non-existant) technical skills, Apache is probably the best choice.
If you need to be able to scale up a large web application quickly, and if you have the technical skills to install, configure, and manage your web server yourself (or if you have a SysAdmin to do it), you may want to think about LiteSpeed or Nginx.
If you are using the .NET framework, or other proprietary Microsoft tools in a Windows server environment, you'll probably be using IIS.

If you are using a typical shared hosting or even most VPS plans, you should almost never need to restart your Apache web server.
If you do have a problem that cannot be solved any other way — and you are not already actively managing and configuring your web server — your best bet is usually to call customer service.
If you are actively managing your web server from an SSH terminal, there are several different ways to restart Apache. The easiest, safest, way to do it is to restart gracefully:

apachectl graceful

(You may have to do sudo apachectl graceful.)
If this doesn't work, try:

apachectl restart

Most shared hosting providers have mod_rewrite enabled by default. If yours doesn't, or if you are having problems with it, the easiest thing to do is contact your hosting company's customer support.

Nginx is a web server.
Nginx is free and open source, and is one of the more popular web servers in common use. Many people see it as a strong alternative to the more commonly used Apache web server.
Incidentally, Nginx is pronounced like "engine-ex."

No.
Nginx and Apache are both web servers. Using both would be redundant.

IIS is the Windows web server.