An administrator is typically responsible for the declarative (point and click) configuration changes of the system. 

1 -For a company that has approximately 50 users, one administrator should be fine.  My organization has about 300 end users on one particular instance of Salesforce, and I am the only administrator/developer.  It's always helpful to have someone around who is knowledgable about the system that you can learn from, but you can definitely learn as you go.  My recommendation is that if you don't know something, always ask on here, or experiment within a Sandbox environment (this is a very hands on approach to learning).

2-Generally speaking, if a business requirement (i.e. tracking a particular metric, implementing an approval process, lead scoring, etc.) falls under the Salesforce environment, it is generally expected that the administrator will take care of ALL of the tasks.  Then again every organization is different, so your company may have a different approach.  

3-I've been supporting Salesforce.com environments for several years now, and I've never had a full-time position where I had to give 24/7 support (and this was with demanding customers).  One great thing about Salesforce is that since it is hosted in the cloud, you never have to worry about maintaining servers or performing system upgrades.  Salesforce takes care of these time consuming activities which generally require the 24/7 support you're referring to.  However if you're doing a new application roll-out, or going through a massive data change, these activities are highly recommended to be done when you do not have any users on the system (typically before or after business hours).

4-It really depends on the organization.  Salesforce admins typically have a cross-departmental role within a company.  They generally represent the sales side of a business (or service depending on whether your using the sales or service cloud of Salesforce), but since the application is extremely flexible, you will often be working closely with other departments.  More often than not, I've seen the Salesforce admin role classified as a "Sales Operations" role, in that you need to work closely with many different departments (customer service, sales, marketing, finance).  In the past I have primarily reported to Sales Operations Directors & VP's, but now I report up to Senior Operations Manager / VP of Operations.  It definitely depends on your organization.

5-Since applications extend the standard functionality of Salesforce, it really depends.  If you're installing an application that will affect end-users, you would need to determine what group of end users are affected and make sure with their manager that it is okay.  Applications that are designed to strictly help out administrators generally wouldn't need permission to be installed, since they aren't affecting end-users.

If you have any other questions please feel free to ask, you'll find that these discussion boards are generally very helpful.