Using over-rested pizza dough as old dough
Primarily what is happening when you add an old dough/pre-ferment/pâté-fermentee to a fresh lot of flour etc is that you are inoculating it with yeast (see definition 1b). This is the equivalent of adding a pre-activated yeast (e.g. sponge) to your flour with some differences. It is also a more direct equivalent of using a biga or poolish. The reason why bakers in former days used to keep some dough over from the day before and use that to start the next day's batch is that they hadn't developed the technology to grow and store the yeast as we do now. The technique is still as legitimate today as it was then.
The major difference is that the yeast in the old dough has used up some of the sugars and other nutrients in the old dough and will have slowed replication as they run out of nutrients (for those interested this follows a classic S shaped/logistic growth curve), whereas in a sponge or other pre-activation there is still a surfeit of nutrients, so the yeast are actively and rapidly dividing. I doubt that the nutrients will be entirely depleted from the old dough, and this would depend on a huge range of factors as to how depleted they actually were, and without specialist chemical analysis you won't be able to tell. As a positive, this depletion of the nutrients means that the yeast have been metabolizing these compounds and releasing the subsequent metabolites as interesting flavour compounds, adding extra taste to your new dough.
The nutrients may be somewhat lacking in your fresh dough if you mix in a substantial amount of the old dough with your fresh ingredients, but in general this won't affect the dough formation as the yeast from the old dough start to grow on the fresh nutrients. You can mix in up to 1:1 ratio of old:new(see page 97; this is the English translation of Clavel's &quot;Le Gout du Pain&quot;) and still make good dough. It is worth noting that most recipes that are old dough based also add some fresh yeast. The addition of fresh is not entirely needed, but will make the rise quicker as the inoculation will be larger and the growth lag shorter as a consequence.
How you store the old dough may affect how useful it is as a starter - If you could freeze, it would store almost indefinitely. In the fridge for several days, maybe a week or two. On the bench, a day or two at best. The colder you can make it, the better it will store, as the yeast will slow metabolism and growth (halted entirely if frozen), and so prevent them from dying.