Not exactly.
POSIX is a standard that specifies how a computer operating system should interact with other software. It's based on the Unix OS, which is the ur-example of this interaction behavior.
Linux is an independent project to produce a POSIX-compliant OS. The different flavors of Unix all start with a basic Unix kernel, and extend from there. Linux was written from scratch to achieve the same POSIX result.
No. Not only are a lot of servers actually running Linux, but there's also a growing amount of Windows, and probably some other esoteric stuff that's not any of those three.
It also depends on what you mean by "internet server". Strictly speaking, the Internet is the underlying information transmission infrastructure. On top of that, there's the computers using that infrastructure to serve content or provide other services to humans and other computers. This upper layer of content and services is, broadly speaking, the World Wide Web, or "web". A lot of "internet" servers are actually "web" servers. The server that runs this forum, for example, is a webserver.
On the other hand, the industrial-strength routers that move information around along one data path or another are also computers. Imagine an army of tiny switchboard operators, sitting at the junction of every major and minor information pathway, flipping bits this way and that to keep the messages flowing. These can be thought of as "internet servers". They can run Unix or Linux, but a lot of them run custom-built, task-optimized operating systems like Cisco's IOS.
The forward slash / is a Unix convention widely copied, as in Linux. IIRC, Microsoft uses the backslash \ because they'd already decided to use the forward slash as a special character in their DOS command line utility.
But nowadays things can get pretty weird. URLs use the forward slash as a path delimiter, as in
http://internationalskeptics.com/forums. As a result, Microsoft's webserver app, IIS, is perfectly comfortable using the forward slash to resolve URLs (even if it has to translate them to backslashes when looking up the local resource behind the scenes; I'm not sure if that's even true).