Check this article, might help you to understand the system.

RuO2 is one of the best OER catalysts there is, however, in case of nonaqueous solvents the situation is very different as there is no water. And since Li2O is the only oxygen carrying content I think 2 Li2O ---> 4 Li+ + O2 +4e- is probably the anodic half cell reaction. For ORR also water might be formed if you get protons from somewhere, but obviously the electrolyte is dry and aprotic so what else could be the (total) half cell reactions?

But oxygen evolution is always a multistep process and there is loads of possible intermediates. (oxygen related ions and compounds).

It is always good to use a reference electrode to find out such, although even then it is not so easy to decide what is the reaction. Best is probably charge and discharge and identify the products. OER and ORR are so irreversible that I would say you cannot easily make the conclusion about the reactions happening on the basis of nearness to E0. There is too much noise of other reactions at OCP. It was very difficult to measure the E0 of 2 H2O O2 + 4H+ +4e- (extrapolation from two branches, anodic and cathodic on Pt)

Some thoughts , but this is just a quick thing, ... is that a) is Li2O a conducttor? And what is the consequence of that

b) Is O(2-) soluble in the solution? if yes then maybe Ru catalyses 2 O(2-) --> O2(aq) +2e- (both directions, multistep mechanism) and Li2O 2 Li+ + O(2-), solubility related???

I think you could even have local electron transfer from the Li2O surface to the Ru surface meaning if a Li+ leaves to the solution the electron might be transferred to the Ru and reduce oxygen there and such alike

Typical electrolyte degradation products are lithium carbonate, lithium hydroxide, lithium formate or acetate. The one that is probably easiest to identify is lithium carbonate. The formation of Li2CO3 results in sharp increase in charge overpotential. If you can let the cathode /electrolyte sufficiently degrade,  you can observe its formation through XRD. FTIR / Raman are also effective and simple tools for lithium carbonate detection. 

More complex techniques involve running GC-MS of electrolyte from the cell or even running NMR analysis. 

Personally, I've utilized XRD to detect electrolyte degradation. I've also utilized GC-MS with good results (although unpublished) 

You can refer my paper if you want to see how I've used XRD - 

https://www.researchgate.net/profile/Yongzhu_Fu/publication/299359231_Chemically_Synthesized_Lithium_Peroxide_Composite_Cathode_for_Closed_System_Li-O2_Batteries/links/57361a9b08ae9f741b29ce09.pdf

You can contact me if you have any questions on using GC-MS for this. 

Hope this helps. 

Article Chemically Synthesized Lithium Peroxide Composite Cathode fo...

Thank you very much, Dr. Sujoy Sarkar, Dr. Sönke Schmachtel and Dr. Amruth Bhargav for your suggestions!!!