The cited reaction of Copper metal and dilute Nitric acid has long known to produce ${NO}$ gas, although the precise reaction mechanism remains complex.
The mechanics, in my opinion, likely involves a REDOX pathway with, for example, ${NO2}$ as a radical product in the case of concentrated Nitric acid proceeding, in this case, somewhat simply as follows:
$\ce{Cu + 2 HNO3 -> Cu++ + 2 OH- + 2 .NO2}$
and, with more acid, Copper nitrate, also a cited product, with the net reaction now being completely in agreement with noted Reaction R2 above.
Perhaps, less complex is to quote Wikipedia on the mechanics of the interaction between Nitric oxide and oxygen:
When exposed to oxygen, nitric oxide converts into nitrogen dioxide:
$\ce{2 .NO + O2 → 2 .NO2}$
This conversion has been speculated as occurring via the ONOONO intermediate.
Now, I actually agree with the speculation that ${N2O4}$ is likely of assistance here, but how?
To answer, I refer to another educational source, which again repeats, albeit, noting the improbability of the termolecular pathway, quoting:
An elementary termolecular reaction involves the simultaneous collision of three atoms, molecules, or ions. Termolecular elementary reactions are uncommon because the probability of three particles colliding simultaneously is less than one one-thousandth of the probability of two particles colliding. There are, however, a few established termolecular elementary reactions. The reaction of nitric oxide with oxygen appears to involve termolecular steps:
$\ce{2 .NO + O2 → 2 .NO2}$
However, of import on the speculated pathway, to continue quoting:
An elementary reaction is at equilibrium when it proceeds in both the forward and reverse directions at equal rates. Consider the dimerization of ${NO}$ to termolecular, with ${k_1}$ used to represent the rate constant of the forward reaction and ${k_{-1}}$ used to represent the rate constant of the reverse reaction:
$\ce{.NO + .NO ⇋ N2O2}$
So, my opinion as to the speculated pathway that avoids the unlikely, but widely promulgated termolecular reaction (which may instead involve ${N2O4}$) is as follows:
$\ce{.NO + .NO ⇋ N2O2}$ (fast equilibrium)
$\ce{N2O2 + O2 -> N2O4 (slow)}$ Source here
$\ce{N2O4 ⇋ .NO2 + .NO2}$
where the net reaction is in accord with the more widely claimed single step reaction:
$\ce{2 .NO + O2 → 2 .NO2}$
implying, I would argue, only apparently speculatively (as there is a confirming source) that there is no direct action of oxygen on the ${NO}$ radical absence water (in complete contravention to the widely promulgated reaction above), but relatedly per a cited reference, instead on its possible self-reaction non-radical product ${N2O2}$, which then forms the non-radical intermediate ${N2O4}$, being probabilistically more likely the precise path.
Those doubting this comment are welcomed to review the associated biochemistry of the ${NO}$ radical in the human body as, for example, presented in this 2015 article Breathing new life into nitric oxide signaling: A brief overview of the interplay between oxygen and nitric oxide☆.