If you are in a windows app or a console app there is by definition NO
HttpContext. HttpContext is only available when you make a request to an
IIS server for a web page. Outside of that request there is no context.
Depending on what you want to get from the context there are other ways to
instantiate SharePoint objects. In a windows or console app you need to use
them. Also, the Page Load event of a web page is not something you can
call. It is raised by the IIS server when the page is loaded.

Tell us what are you trying to do and we can try to help.

I want to get all the output
cache values using the following syntax at the level of every site
collection.

HttpContext.Current.Cache

and then remove the cache for an individual item by passing its key with the
following syntax.

HttpContext.Current.Cache.Remove(<key>)

I can't use this in a console / windows / web application. So I thought I
would put this in a custom application page and would call this page through
code from a console / windows application. Also, this needs to be executed
for every load balanced server in the farm.

As I've already said. If the cache is properly configured you shouldn't
need to invalidate it when you push new content. Rather than spending a
lot of time trying to jump through hoops to invalidate the cache why don't
you look at how its configured and get it configured correctly. Then the
extra programming won't be necessary.

This a shot in the dark here, but it sounds to me like you should really
be extracting whatever code the you have in that Application Page to
some other module so that it can be called from multiple places...

Go grab the file from the server and mimic its functionality on
your own page...I like it!

you sure you don't work for Microsoft technical support? Your questions
are always very detailed and have nuances that are rarely encountered on a
day-to-day basis...yet you seem to have them a lot. You must have some really
large-scale projects with a bunch of developers?