Fundamentals of Li-ion Functionality
Based on the Wikipedia article, I am inclined towards thinking that the lithium on the electrodes is neutral. The lithium ions move through the electrolyte to facilitate the two half-reactions. The electrons in the half reaction participate in closing the external circuit, so the lithium ions and electrons are moving in the same direction, except one does so inside the battery and the other outside the battery).

Onto your questions:


The lithium in the anode is neutral, not an ion.
There's no traditional bond per se, but it's stored in the energy difference between $\mathrm{LiC}_{6} + \mathrm{CoO}_{2}$ versus $\mathrm{C}_{6} + \mathrm{LiCoO}_{2}$. There is an interaction here, but I wouldn't call it a "bond."
Well, the anode or cathode isn't more energetic, that's not quite the right term. The $\mathrm{LiC}_{6}$ is more energetic than other species. It just happens that this happens to be the anode of a charged battery. Once the battery discharged, the anode is still that side, but it doesn't have more energy.
Electrical energy doesn't work that way per se. We generally say we build up a potential difference $V = \frac{\Delta E}{q}$. The extraction of energy from this difference is dependent on the circuit. If you have an incandescent light bulb, the kinetic energy of the traveling electron is converted to heat and light, for example.
When the battery is discharged, the lithium atoms have moved to the metal oxide. That's the thermodynamically more stable of the two substrates for the lithium to be adsorbed onto.


I've never looked this up before, but this is actually reall cool. We'll basically using the thermodynamics of lithium preferring to be adsorbed onto one electrode over the other. In order for the lithium atom to get there, it has to move as an ion through the electrolyte and the missing electron moves through the external circuit. I think your confusion is here: the system wants to move the lithium atom, so we'll extract something useful out of it by forcing it to move the lithium ion and the electron separately.