Pros and Cons of running a deck with no 1-mana spells?
When I'm playing standard, the game is typically decided in the first 5 or 6 turns. Consider how devistating a Titan was, and the number of extremely powerful 6 drops compared to extremely powerful 5 drops. Now in the current set, development actively tried to extend the game further out, but still in many cases the game is set in stone (barring a true bomb, of course) by turn 6. If you don't have a solid 1 drop in a standard game, you aren't so doomed that you scoop right away, but you are at a significant disadvantage because you lost the ability to play on one of the 5 game defining turns.

When I'm playing draft, assuming the opponent doesn't get mana screwed, mana flooded, color screwed, or some other 'lame' win, I expect the game to really not be decided until turn 8 or 9. And due to the sporadic nature of bombs in your draft decks, can completely turn around at any time. So missing out on a first turn creature and falling a little behind isn't a big deal. It's still not good to start at a disadvantage, but you're much more likely to come back from this in draft than in standard.

Now in something like Commander, when you have 40 life, a 1 mana creature doesn't do much at all. In fact, I only have 4 one mana spells at all (which, drawing 7 cards out of 100, means I have a about a 1/3 chance of getting one - not very high) in my main Commander deck, and none of them are creatures. They are all highly utility too: Sol Ring, which i consider more like a land than a spell, Shrieking Affliction, which isn't much of a turn 1 spell, but does a lot of damage over the course of the game, Virulent Swipe just because death touch is very important to the deck, and Quest for the Gravelord to profit from people fighting eachother. So really, only one of those cards is intended to be played on turn 1, and is still viable at any time in the game. (5/5s are always welcome!)

So, it's important to know what the tempo of the game is. I have control decks that don't mind not having a 1-drop. Multiplayer doesn't normally need one. It's all about knowing your deck, the tempo, and even your opponents. Like so many things... playtest, playtest, playtest!!