IOS – Is there a way to see a list of all the apps available on the app store

This is now possible since iOS 4.3.3 by going into the iPhone App Store > Updates > Purchased. It provides a list of your iPhone app purchases and allows redownloading from the iCloud.

Also you can visit reportaproblem.apple.com and all recent purchases will be shown.

After a bit of research (in the stores themselves, using Google, and in the Apple Developer Program resources), I'm rather sure no list of these is available from Apple themselves or from any third party.

I found the best ways to find the apps tagged as "Essentials," is to look in these places:

• Both the Mac App Store and the iTunes App Store have a link on their front page called "New to the (Mac) App Store?" Most apps listed there, which are not games, have the "Essentials" badge. It's not the way to find them all though, as e.g. the WolframAlpha app you mentioned is not listed on that page in the iTunes Store. There are also some apps which don't have the badge, e.g. Shazam is tagged as "Editors' Choice," but not "Essentials" in the Mac App Store.
• The iTunes App Store also has a link "App Collections." Although the page this link leads to is titled "Essentials App Collections," quite a number of the apps listed in the collections don't have the "Essentials" badge. On the other hand, you can find a lot of "Essentials" this way.

My impression is the difference between the "Essentials" tag and e.g. "Editors' Choice" is not very well defined, and just picked by a specific Apple store editor. And I noticed some of those apps may be only essential to a very specific crowd. E.g. the djay 2 app, also tagged "Essentials," is most definitely the best and most popular DJ app for the iPad out there, but it's only essential to a select group.

In the end, although there is no definitive list, going though the various collections made by Apple editors, will lead you to a lot of "Essential" apps.