Does Efficiency apply to breaking melons?

In 1.4.7, I was able to enchant a pickaxe with Efficiency and break melons quickly with it. Now in 1.5.2, I'm not seeing the same behavior.

In the latest version, is there still a way to break melons more quickly?

In 1.5, the way tool enchantments works was changed so that Efficiency and Fortune only work if the right tool for the block is used. The "right" tool for pumpkins is the axe (for some reason), but melons have no "right" tool and therefore default to the sword.

To answer this question with science I set up a flat world in Creative Mode on 1.5, set up a pile of melons and some tools with and without enchantments, and then switched into Survival Mode to see how long they each took to break a certain number of melons.

With a normal diamond sword, it took me 16.9 seconds to break 13 melons in a line. With an Efficiency V diamond sword, it took me 5.4 seconds to break 13 melons in a line. Other tools are in this table:

Sword        16.9s
Ench. Sword   5.4s
Ench. Pick   23.2s
Ench. Axe    23.2s

As you can see, Efficiency V on picks and axes actually breaks melons slower than an unenchanted sword. However, this only demonstrates that swords are the right tool for melons – you can't actually enchant swords with Efficiency in Survival mode.

I did a similar test with Fortune III, comparing normal diamond swords, pickaxes, and axes with enchanted counterparts, plus one sword with Looting for completeness, and counting how many melon slices were dropped, and the results were inconclusive:

Sword         35 melon slices
Ench. Sword   38
Pick          27
Ench. Pick    40
Axe           25
Ench. Axe     44
Looting Sword 26

The variation between the tools and their enchanted versions is not statistically greater than the variation between the different types of unenchanted tools, which indicates that Fortune has no effect on melons in 1.5.2.

Conclusion

The fastest tool for breaking melons normally available in Survival is an unenchanted diamond sword. If you can somehow manage to create one though, an Efficiency sword is the fastest tool possible. The non-tool way to break melons is using pistons, which "pops" them instantly, but this requires finding redstone first, and ideally would be automated with more redstone, making pistons a later-game method of quickly harvesting melons.

I use pistons to break up melons. Occurs almost instantly.

As far as I know, enchantments now only apply to the tool when breaking or interacting with something that the tool is normally used for. Ex, Breaking cobwebs with a sword.

To answer your second question, the tool that breaks melon blocks the quickest is a sword. However, it damages it by -2, meaning double the damage.

You may collect melon blocks by using silk touch enchanted axe or shears, also.