Minecraft – Do calves jump

Read the long explanation of my answer below, but here is a chart that summarizes how big of a gap you can jump along with how far you must fall to make the jump and the damage you will take (without mitigation). You must make a full running jump for this chart to apply:

Gap Width (Blocks)  Blocks fallen below jump level  Damage taken from fall
2                   none                            none
3                   2½                              1
4                   7½                              3½
5                   14½                             7
6                   23½                             11½ (death)

So without falling damage mitigation you can make a jump over a gap that is at most 5 blocks wide, after falling about 14 blocks down from the point where you make a running jump.

I will test this in-game at a later time to make sure this is realistic.

So this answer only works given a number of assumptions hold.

First assumption, Minecraft models gravity using a parabolic equation of the form y = a(x - h)^2 + k, where x is the size of the gap you want to jump and y is the height difference you will be after clearing the gap (negative indicates you are below the point you jumped). a is a tunable parameter that describes the rate at which you will fall (related to the constant of gravity). k is the highest you can jump from the ground in Minecraft and h is the horizontal distance you will have cleared while jumping that high after running forward.

Second assumption, that k (the height you can jump from the ground) is 1½ blocks and h (the horizontal distance you will have cleared at that height) is 1 block.

Third assumption, that horizontal momentum is preserved. Basically this means that you will travel at the same speed horizontally throughout the jump.

These three assumptions give the equation y = -(x - 1)^2 + 1½.

So let's say you want to jump a gap that is 2 blocks wide. To see how far down you must fall before making that jump, plug x = 2 into the equation and you get y = ½. This corresponds with being able to safely jump a gap that is 2 blocks wide with a running jump, and you will reach a block on the other side of the gap while being a little higher than ground level.

Falling damage is calculated as (number of blocks fallen x ½) - 1½. So, let's say you want to jump a gap that is 5 blocks wide. Plug x = 5 into the equation and you get y = -14½. This means you will reach jump a gap that is 5 blocks wide after falling 14½ blocks and taking about 7 hearts of damage.

Sprinting Added

You can jump significantly further with sprinting; at least three blocks horizontally. This answer assumes no sprinting but I will try to update it ASAP.

I always put my lip just high enough so that I don't bump my head on it (that is, with 2 blocks of air between the ground and the lip). I have watched it stop spiders and have never encountered a spider above the lip. James' answer is correct about the spider AI.