Help Finding App -- Download & Save files?

The obvious choice would be one of the "Quickoffice" apps. There are alternatives - read the descriptions in iTunes. Just search there for "quickoffice".

You cannot download them into a file as the iPhone does not have a user accessible file system. You can only do what you wish via an app that handles those kinds of files. For office files, I have tried both QuickOffice and Docs2Go and prefer Docs2Go which has a free desktop app to sync files between the iPhone and your computer. QuickOffice uses a web interface for connection on the same subnet as your computer (you activate it on the iPhone, then point your computer browser to the IP address).

These apps, like all iPhone apps, do not share their files with other apps though - you can use them to open and edit word/excel files and upload/download them to your computer, but they store the files on your iPhone in their own application, not some generic file folder.

I use Files to download and store excel, word, pdf, and other files. It works over wifi or charging cable.
There is a free version "Files Lite" you can try before you buy.
If you save files on the free version and then buy the full version, the files do not transfer over, you will have to re load them.
Excellent app. I use it daily with no problems.