Possible side reactions in electrolysis of concentrated NaCl

This is a very basic question about which side reactions can take place if the electrodes are not shielded from each other during electrolysis of concentrated $\ce{NaCl}$.


So, this is the desired net reaction for this process:
$$2\ce{NaCl(l) + 2H2O -> 2NaOH(l) + H2(g) + Cl2(g)}$$


At the anode, oxidation of chloride ions to chlorine gas: 
$$\ce{2Cl- -> Cl2 + 2e-}$$
At the cathode, reduction of water to form hydrogen gas and hydroxide ions:
$$\ce{2H2O + 2e- -> H2 + 2OH-}$$


The section '*The need to keep all the products separate*' on this [page](https://www.chemguide.co.uk/inorganic/group7/diaphragmcell.html) gives $ 2$ by-products that are formed when $\ce{ H2}$ and $\ce{Cl2}$ stay in the system:
$$\ce{H2 + Cl2 -> 2HCl}$$
$$\ce{Cl2 + 2NaOH ->NaOCl + NaCl + H2}$$


I wonder if $ 2 $ additional side reactions can take place, simply with the gases diffusing to the other electrode, resulting in the reverse redox reactions:


At the cathode, reduction of chlorine gas to chloride ions: 
$$\ce{Cl2 + 2e- -> 2Cl-}$$
At the anode, oxidation of hydrogen gas to form hydronium ions:
$$\ce{H2 + 2H2O -> 2H3O+ + 2e-}$$


1. Aren't these reverse (and spontaneous, based on the standard potential of these reactions) redox reactions also likely to happen when the electrodes aren't shielded?
2. Which one of all these side reactions (or maybe others?) are more likely to occur in this scenario? I think $\ce{ H2 + Cl2 -> 2HCl}$ as both are in gas form and are likely to encounter each other


Thanks a lot.


