All areas necessary in Dark Souls?

When I first started the game, I followed the same path as many other beginners and went to the first bell and then to the Depths, Blight Town and to Queelags Domain to ring the second bell.

However, I really found Blight Town much earlier since I went from the first bell to the Darkroot Gardens and to the "exit" of Blight Town running past the drakes in Drake Valley and as far as I can see, there really isn't any need to go all the way through the depths and most of Blight Town at all?

Heck, if I have the Master Key I don't even need to go through the Darkroot Gardens. I can just go to New Londo and straight to the second bell skipping also the gardens.

My question:

Is it necessary to visit all areas to progress through the game or are some just there to provide loot, NPC:s (with their story elements) and heaps of practice and souls?

As you can tell I just rang the second bell and haven't progressed much further than this.

I did kill the Moonlight Butterfly and really couldn't see any need to do this either as it didn't bring me any closer to finishing the main game.

Please advice! :-)

PS: I AM actually going to play through every single square inch of this wonderful game but it intrigues me that some areas and bosses just feel totally unnecessary... :DS

The Wikidot Dark Souls Wiki lists 4 areas as "optional" - The Great Hollow, Ash Lake and Painted World of Ariamis on the Areas page, plus Valley of Drakes on its own page. (I'd argue Darkroot Basin is also optional, but the border between Darkroot Garden and Darkroot Basin is somewhat blurred.)

I'm unsure of the role of the Moonlight Butterfly in the game progression, but there are definitely other parts of Darkroot Garden that need to be completed before the game can be finished.

I haven't finished the game yet myself (though I am close), but since I did not choose the Master Key as my starting gift, I have had to unlock each area in sequence. The Master Key does allow you more freedom to access areas out of sequence, but I think the game is better structured without it.

With clever use of the Master Key, you can skip directly to the Bell Gargoyles & Quelaag without fighting any other bosses. You could skip Taurus Demon, Most of the Undead Parish, Moonlight Butterfly, Lower Undead Burg, Capra Demon, the Depths, Gaping Dragon, and Most of Blightown.

From Blighttown you can get to the Great Hollow and from there to Ash Lake, but it's rather easy to miss, I didn't even realize they existed until a friend told me after I had finished my first playthrough.

In one of the final areas of the game is another shortcut that can be opened through leveling of a covenant relationship that allows you to bypass most of the area as well as two bosses.

So it definitely isn't necessary to hit all of the area or even all of the bosses to complete the game. By my count, you can get through the game fighting only 13 of the 21 bosses.

EDIT: Without the Master Key you'll still need to fight the Taurus Demon but then, as O.T.2 pointed out in the comments, from there you can go through Darkroot Garden/Basin to the Valley of the Drakes and then to Blightown, skipping Lower Undeadburg, the Depths, and most of Blightown. The Master Key allows you to open the New Londo <-> Valley of the Drakes shortcut, which allows you to skip the Taurus Demon.