I hope you appreciate I'm writing this response not as a definitive answer but as an attempt to show you the actual size of the question you're asking. It's like you've asked for a glass of water but the only way to give you a useful reply is to dump the entire ocean on your head.
You wouldn't ask "What is a developer", hopefully, because you'd appreciate there's a big difference between developing a small web app (as important and valuable a skill as that is) and being the lead code architect for Windows or Linux or IBM DB2. So it is here.
On with the answers
1. What are the security requirements of a server?
To a certain degree this isn't a technical question because the answer is defined by the client/employer; both what they are and what they want and what they can afford (at the risk of stating the obvious a bank has different security requirements from a chain of restaurants). Whole books have been written about this first question alone, so there's a limit to what you can reasonably get out of a question here, perhaps.
Do they value security over getting stuff done in a hurry? Do they handle data or are they in a industry that places regulatory requirements on how they secure servers, services and data?
At a bare minimum, you'd probably want a baseline for how servers are configured, a framework for patching them and some kind of plan for both incident response and disaster recovery in the event an incident proves serious. And securing data, securing a server and securing services are different things that need to be considered both separately and together.
Just to be clear, a service might be something like your web site, or email. This can be provided by a number of servers, both in terms of distributing workload and providing redundancy.
2. What are the key responsibilities in a system admin's job description?
How long is a piece of string...
Typically a IT department is responsible for providing a bunch of services, and in a well organised department this is done using service level agreements; formal documents that describe what a service consists of, the expected availability and procedures and timings that come into play if the availability isn't there.
System Administrators work to ensure some (or all in smaller organisations) of the IT team's obligations for meeting that SLA are achieved. There may also be elements of R&D/planning, and so on added on but by and large that's what it boils down to.
Even in places without a formal SLA, there will be a bunch of services that is provided to the business by IT and its your job to keep them running. That's pretty much it.
3. What are some of the day to day tasks of a system admin?
In terms of day to day tasks... It's largely defined by the business again, there isn't a common professional definition of "system administrator". To a large degree we're the "almighty janitor" of the IT industry and that means we have to do stuff that others don't want to do and sweep up after others, so it really does depend on the job. I've seen system administrator jobs that include programming, jobs that include managing the phone system, jobs that include none of those but something else odd... it really does depend.
Monitor servers and services to ensure SLAs are being met, and take action to correct things if not. This may require working with a helpdesk team (and in smaller businesses will probably include being the helpdesk too).
There's a reactive element to this ("hey the CEO's email ain't working, find out why and fix it") and a proactive element ("hey the disk storage performance on the email server is appalling, better find out why and fix it before the CEO's email stops working").
Besides that, commonly scheduled maintenance tasks might include checking for physical problems with servers / server rooms, checking logs to ensure systems are performing well and appear secure, planning and carrying out planned changes. And mediocre stuff like making sure the backups went ok, tapes got changed in the backup system, etc.
4. What is the average monthly salary of a system admin? Note: I will be working inside a Windows environment. But your replies do not necessarily need to be constricted to a Windows environment.
That's very much a local market conditions question so no one here can answer it - a "worldwide" average is no help to you because it will include people in New York, Bangalore and London (as well as lots of other places) and those 3 cities alone have vastly different salary expectations. It varies based on location and industry, as well as obvious stuff like experience and qualifications, so even if you happened to live in the house across the street from mine my answer would still have to be "it depends".