Minecraft – Will grass grow from just a dirt field

With the silk touch enchantment on any tool, you can obtain a 'grass block' when digging instead of a dirt block. Place the grass in a patch of dirt and it will spread normally.

This is also helpful if you want to transplant Mycelium blocks.

Source: http://www.minecraftwiki.net/wiki/Enchanting

Yes, making a bridge more than one block wide will increase the speed of the grass spreading. Grass can spread to any adjacent blocks (including diagonals), from one level above to three levels below the "source" block. If you were to build a one-wide bridge, the grass would have to spread from the first block in the bridge to the second, then to the third, etc. However, with a > 1-block-wide bridge, there are multiple blocks that this algorithm could be run on. Also, grass could spread to the block in front or the blocks diagonally from it, allowing for the grass to propagate faster.