Minecraft – Do different types of meat fill more hunger bars

Yes, food now only replenishes your hunger bar, and if the hunger bar is high enough, you will regenerate health at a fixed rate (~1/2 hp/5 sec at 90-100% sated). Conversely, and depending on difficulty, when you're starving you will either lose health until you hit 50% (easy), 5% (normal) or die (hard).

Various actions including walking, swimming, running, and stabbing things will increase your hunger (though notably not boating).

Food now stacks to 64, so it should be plenty easy to carry around enough for quite a while.

If you don't want to die while fighting hostile mobs, don't be so reckless; wear armor and/or take them out at range with a bow. As of 1.9 and 1.0 release, you can also use potions (Potion of Instant Health and Potion of Regeneration) to supplement your longevity. A Golden Apple also gives the same effect as the Potion of Regeneration, though owing to the extreme rarity probably not worth it.

The food level increases first. That's all there is to it. Here's a piece of code from FoodStats.java, courtesy of the Minecraft Coder Pack:

public void addFoodStats(int foodIncrease, float saturationIncrease)
{
    foodLevel = Math.min(foodIncrease + foodLevel, 20);
    foodSaturationLevel = Math.min(foodSaturationLevel + (float)foodIncrease * saturationIncrease * 2.0F, foodLevel);
}

As you can see, the food level is increased first. Then, the saturation level is added based on the new food level value.

Therefore, waiting vs constant top-offs will probably have little effect on your food's efficiency.