Does gravity play a role in the Earth's equatorial bulge? I'm trying to understand why the Earth bulges at the equator. But before looking at the Earth, which introduces gravity, I wanted to make sure I understood the shape of some rotating objects and if/why they have a bulge.
A metal sphere: A very rigid object. If this is rotated then there is no bulge because it requires a lot of energy to deform the object.
A ball of sand: Nothing is connected. If this rotated then all the sand is simply thrown off due to inertia, leaving nothing behind after a time.
A ball of dirt: Somewhat rigid. The dirt wants to "fly off" due to inertia but tension keeps the object pulled together, as a result there is bulge at the equator - where the velocity is fastest and thus more "tension" (more bulge) is needed. 
Does the "ball of dirt" situation accurately resemble Earth? If so, then does this not suffice to explain why the Earth bulges without even needing to mention gravity? Or is Earth more like the "ball of sand" which is can only be kept together (while rotating) under the force of gravity? Or is it something between or neither at all?
I'm afraid I'm terribly confused as to how the various effects come in to play and interact to create an equatorial bulge. Some sources have tried to explain in terms of an energy equilibrium but I'm not entirely satisfied with that as an explanation (actually that source even indicates the equilibrium approach is not so straightforward). 
 A: 
Does gravity play a role in the Earth's equatorial bulge?

Absolutely.

A metal sphere: A very rigid object. If this is rotated then there is no bulge because it requires a lot of energy to deform the object.

Of your three models, this is the closest to the truth. Metal is not quite as rigid as you think. It compresses under pressure, bends under strain. Solid metal can be quite ductile and malleable, particularly when it's warm.
Most of the Earth is solid rock. It, too, is ductile and malleable.
That ductility and malleability, combined with gravitation, is all that is needed to explain the Earth's equatorial bulge. The Earth's self-gravitation forces the Earth to take on a shape whose surface is very close to an equipotential surface. The equipotential surface for a rotating object is not spherical. It is instead ellipsoidal. In other words, an object with an equatorial bulge.
A: The inside of the earth is still, partly liquid, source: Mantle  So, it's very viscous, slow moving under very high pressure, but it's still liquid enough and the Earth's crust is really quite thin and not nearly rigid enough to resist forces from below.
Taking the earth as a whole, I'm not sure there's a really good single single analogy.

A metal sphere: A very rigid object. If this is rotated then there is
  no bulge because it requires a lot of energy to deform the object.

OK, I can't do the math here, but in a nutshell, an Earth made of Iron and an earth made of water would have the same bulge - er, almost certainly.    Certainly a hot earth made of iron would have the same bulge as the earth does now, cause the inside would be liquid (and the formation of planets tends to get them very very hot).   The bulge has nothing to do with density or tension.  Just angular momentum vs gravity.  An earth made of fluffy marshmallow vs an earth made of iron would have very different gravity but the corresponding angular momentum would be equivalent to the ratio in gravity so the bulge would be the same no matter what the material.
Now, if you could construct an entire earth - cold and solid all the way through, made of Iron into a perfect sphere, maybe such an earth could resist the centrifugal forces that makes planets bulge. - MAYBE, but it would have to be artificially constructed and even then, I'm not sure it would hold and even if the strong metallic bonds did hold, there would still be stretching around the middle due to rotation anyway.   It's roughly the same principal of how high can you build.   At a certain point, even the strongest metals won't support themselves if you keep building bigger and bigger - so generally speaking, hardness isn't a factor at all in planetary bulges any more than mass is - at least at the size of a planet.   All that matters is the ratio between gravitation and rotation.   At the size of a large asteroid or small moon, then hardness matters, but not for an earth sized planet.  

A ball of sand: Nothing is connected. If this rotated then all the
  sand is simply thrown off due to inertia, leaving nothing behind after
  a time.

You have to compare the gravitational force to the centrifugal force.   If the Earth spun some 18 times faster, sand could fly off the equator, effectively weightless.   The Earth spins much to slow for that to happen though when the moon formed, the Earth may have spun some 5 or 6 times faster than it does today - enough that, at that speed of rotation, you'd feel noticeably a little lighter on the equator, but not enough for stuff to fly off.   in other words, Gravity 42, Inertia 0.  It's not even close.
On Jupiter, which spins faster than any other planet, and assuming it had a hard surface you could stand on, you would notice a measurable difference in your weight at the pole vs on the equator.  Source:   Jupiter
Quote: 

A day on Jupiter is only 9 hours and 48 minutes long. Such fast
  rotation causes Jupiter to be somewhat squashed due to centrifugal
  force: its polar radius is 4000 km less than its equatorial radius
  (the latter being 71,392 km). That means you'd weigh almost 25% more
  at Jupiter's poles than at its equator, that is, if you could find a
  place to stand! So a 100 pound person (on Earth) would weigh 230
  pounds at Jupiter's equator but 285 pounds at the pole!

Now, if you get small enough objects with very low gravity, what you suggest can happen.  There are comets and asteroids that spin fast where anything lose on their rotation flies off, or, effectively negative gravity.   That can't happen with a planet, it can only happen with small objects and even then, it's pretty rare.
Example:  negative gravity asteroid
OK, moving on: 

A ball of dirt: Somewhat rigid. The dirt wants to "fly off" due to
  inertia but tension keeps the object pulled together, as a result
  there is bulge at the equator - where the velocity is fastest and thus
  more "tension" (more bulge) is needed.

As discussed above, inertia would need to be sufficient to effectively make gravity zero.   At the size of a water balloon, gravity makes no sense, and things like surface tension is what you can see.   Small things don't have gravity to speak of.   At the size of a planet, bulges happen, but nothing flies off.   The bulge and "flying off" is the same force, but Gravity wins and it's not even close.

I'm afraid I'm terribly confused as to how the various effects come in
  to play and interact to create an equatorial bulge. Some sources have
  tried to explain in terms of an energy equilibrium but I'm not
  entirely satisfied with that as an explanation (actually that source
  even indicates the equilibrium approach is not so straightforward).

Gravity and the inertia from spinning can seem counter intuitive.  I get that.  The idea that standing on the Earth's equator and knowing that you're spinning at over 1,000 MPH, you might think, moving that fast, maybe I should fly off the earth and for what it's worth, that confused Ptolemy too, that's why he rejected the rotating earth idea that Aristarchus of Samos had proposed.  (Aristarchus followed Aristotle and I sometimes wonder what might have been had Aristotle and Aristarchus met and had that discussion, but Aristarchus was born 12 years after Aristotle died, and by then Aristotle's writings, which were accepted as scientific gospel for nearly 1,800 years, had been finished.   And, I'm not trying to put down Ptolemy or Aristotle, that's probobly not fair.   I only bring up the example to point out that the idea that the Earth was rotating was sufficiently confusing that many of the brightest people on earth doubted it was possible for many centuries.   
So lets use a real world example:
You're in a car and the car takes a turn, or banks, if you like and you feel yourself pulled in the opposite direction of the bank.   That's quite similar to the interaction between gravity and centrifugal force on the surface of the earth.   Now in a car, gravity pulls you down and when the car banks a hard turn - that pulls you sideways, the forces are at a 90 degree angle.   Gravity down stays constant but gravity sideways depends on the speed and tightness of the turn.
On earth, lets say the earth's equator - same thing, except Gravity is down and the centrifugal force, or inertia, or bank - whatever you want to call it, is effectively up, not sideways.
OK, back to the car.   The car is driving very fast, just over 1,000 MPH, but it banks the turn very slowly, in fact, the turn on the wheel is so slight, you barely even see it and it takes 6 hours and over 6,000 miles of driving to complete 1 90 degree turn.   Because the turn takes so long, the force you feel sideways in that car is very slight - about 3 OZ to a 200 lb person.
Same with the earth.  The Inertia, reduces your gravity less than 1/2 of 1 percent - so little, in fact, you don't even feel it.   
hope that wasn't too long.  :-)
