Does relative amount of lithium and cobalt oxide change in recharged battery?
About maximum $20~\%$ (as I remember, perhaps more now) of the $\ce{Li}$ goes (as $\ce{Li+}$) out of the structure of $\ce{LiCoO2}$, through the elctrolyte, and into the electrode on the other side of the battery. That electrode consists mainly of graphite, which intercalates the $\ce{Li}$ in it's layers in a quasi-metallic form.

So it is $\ce{Li_{1-x}CoO2 + x~Li}$ after charging. $\ce{LiCoO2}$ has a special structure that does not break down when some $\ce{Li}$ is missing, except if you keep a charged battery at over $\mathrm{50~^\circ C}$ or so.

By the way, the related wikipedia article tells you all this. Two days of researching? ;-)