You are right that the plasma will match the composition of the air in that it will be mostly N2 but also have some O2 in it.  O2 is pretty reactive in a plasma.  Pure O2 plasmas are used to clean organic contaminants off of silica surfaces for instance.  N2 on its own is stable, but in a plasma is will react quite a bit.  The plasma will likely contain N+, N2+, N3+, and even N4+.  Those species can go on to form ammonia or other reactive molecules or ions.  So, there can still be a number of reactions involving N2 in an air plasma.

Yes, you are right.

The chemistry of air plasma is very complex involving 100s of species and 1000s of reactions (at least theory supports these numbers). A lot depends on the operating conditions of the plasma- the power input to the gas, the flow conditions/mixing, frequency of operation, pressure and so on. Oxygen in gas yields ozone, singlet oxygen, and with nitrogen they form nitrogen oxides. Species from pure nitrogen are already listed in the former answer. The air plasma chemistry will be much more complex in the presence of water molecules (humid air). Like you mentioned, the physical cleaning and surface activation action is primarily due to ion and species bombardment on the surface. The chemical cleaning action is due to the activated molecules that react with organic substances.

For details on air plasma chemistry one can refer to-

Herron JT, Green DS. 2001. Chemical kinetics database and predictive schemes for nonthermal humid air plasma chemistry. Part II. Neutral species reactions. Plasma Chem. Plasma Process. 21, 459-81.

Sieck LW, Heron JT, Green DS. 2000. Chemical Kinetics Database and Predictive Schemes for Humid Air Plasma Chemistry. Part I: Positive Ion–Molecule Reactions. Plasma Chem. Plasma Process. 20, 235-58.

Gaens WV, Bogaerts A. 2013. Kinetic modelling for an atmospheric pressure argon plasma jet in humid air. J. Phys. D: Appl. Phys. 46, 275201.

David, Plasma is one of the four fundamental states of matter and is composed of a gaseous mixture of negatively charged electrons and highly charged positive ions, being created by heating a gas or by subjecting gas to a strong electromagnetic field. However, true plasma production is from the distinct separation of these ions and electrons that produces an electric field, which in turn, produces electric currents and magnetic fields.

plasma is used in cutting electrically conductive materials by means of an accelerated jet of hot plasma. Typical materials cut with a plasma torch include steel, Stainless steel, aluminium, brass and copper, although other conductive metals may be cut as well. Plasma cutting is often used in fabrication shops, automotive repair and restoration, industrial construction

@Joseph: Just to let you know: plasma is NOT a state of matter in the physical sense because there is not thermodynamical state transition involved in ionisation processes. It is also not necessarily a gas as also liquids (e.g. electrolytes) or solid state matter (e.g. conducting metals) can be described by the means of plasma physics.

@Klatu: back to the initial question: I would not recommend to use an air plasma to clean the grid (with the grid, I guess, you mean part of your electron gun inside the TEM, right?) of your TEM at all because reactive species (foremostly oxygen and nitrogen) radicals and ions can severely damage your TEM grid. There won't also be mostly N2 or O2 plasma because if you have a plasma you will immediately form atomic species, ozone, NOx and a vast amount of other species - a lot of those species can also react chemically with the surface of your grid and form oxides or other compounds, which will alter the performance of your grid or ruin it entirely.

I fully agree with Johannes. Even if you limit the appearing species to a needed minimum you will have to adDress species such as radicals, Atoms,  NOx etc.

I also agree with Johannes and Georg. Once you put your air aka N2 / O2 mixture in, you will obtain an atom / molecule mixture also including ionized species depending on pressure and temperature, set aside other energy sources. Main role is played by the different dissociation rates of N2 and O2. An interesting example: Air, a N2 / O2 mixture at Earth surface, becomes mostly a N2 / O mixture in about 200 km altitude in low pressure conditions, even not yet ionized.