Can some particles go through a benzene ring?

Electrons, maybe.

Atoms, not a chance in hell.

On the left, a ball-and-stick model. It looks like it should be easy to fit something pretty large through the middle, like a carbon atom.

On the right, a space-filling model. This indicates how much space is actually occupied by the atoms/electron clouds in benzene, and you can see that there’s no space in the middle of the ring at all.

Generally, a ring has to be at least twelve atoms big to have any open space in the middle.

cyclododecane (12), cyclopentadecane (15), cycloeicosane (20) in space-filling models

If you use alkene $\mathrm{-CH=CH-}$ units rather than alkane $\mathrm{-CH_2CH_2-}$ units, it doesn’t really matter. You still need more than 15 carbons to have any sort of space in the middle of the ring.

cyclotetradecaheptaene, $C_{14}H_{14}$

cyclooctadecanonaene, $C_{18}H_{18}$

We live in a free world so, in my opinion, you can call it whatever you want. You might want to call it Ben, Tom, Kelly, Gigi, or Baby or ‘Snake biting in his own tail’. I really don’t care: your suggested name is pure nonsense as well.

Ok, let me rephrase it: your name will lead to a structure that people will recognize as benzene. But it is not benzene!.

To make it extra complicated. If you would ask a chemist to draw benzene they might draw exactly the same structure.

Benzene is aromatic. The notion of aromaticity is one of the most important concepts in chemistry, and strangely enough, it is very hard to have a good definition of aromaticity. But you could state that aromaticity explains why certain structures are more stable in comparison with their acyclic structural analogs. All of the carbon-carbon bonds of benzene have the same length and the pi electron system is continuous and uniformly distributed. It is, therefore, a bit cumbersome to draw benzene but most of the time we use Kekelé structures or represent it with an inner circle.

The Kekelé structure indeed looks the same as cyclohexa-1,3,5-triene(*) but as explained that's merely because our representation of benzene is sloppy.

Chemists only use the name cyclohexa-1,3,5-triene (*) if we refer to a hypothetical structure without resonance. It does not exist! To make it clear it is not benzene it is often depicted as a skewed structure

So to conclude disregard all current other answers to this question telling you cyclohexa-1,3,5-triene is the systematic or even the IUPAC name of benzene: it is not. IUPAC only has one name for benzene and, in my opinion, one name is more than enough.

(*) I’m using the latest name writing conventions: locants are mentioned just before the functions which they relate.

Benzene is C6H6 and a pheny group is C6H5* (it can either be phenyl anion(C6H5-), the phenyl cation(c6H5+) or the phenyl radical(C6H5.))
Benzene is a stable organic compound, where as a phenyl is an intermediate formed when benzene is subjected to various reaction.

Here is an example, the second step should clearly show you the difference.

Benzene ring is a compound with formula C6H6. It does not have carbons with 3 double bonds; but has resonance structures which have carbons with 3 double bonds.

Imagine the double bonds in C6H6 delocalizing its pi electrons continously to all the carbon atoms in the structure so that the charge of electrons do not get concentrated in a single region, but get distributed to the whole molecule thereby increasing its stability. This feature of the molecule to increase its stability is called Resonance. Hence it would not be right to say Benzene has 3 double bonded carbons; It just has "double bond character" brought to it because of Resonance.

About your second question whether every molecule needs to have a Benzene ring to be termed aromatic or not- the answer is no.
All aromatic compounds need to follow a rule called Huckels rule to be termed aromatic. According to this rule, if the compound contains 4n+2 delocalizable* pi electrons, it can be termed aromatic. (where n=0,1,2,3...) eg:- Benzene has 6 pi electrons that delocalize themselves continously during the process of Resonance. For Benzene, 4n+2=6. Here n takes up the value of 1, hence it can be termed aromatic.

Nitrobenzene is also an aromatic compound. The delocalizable* electrons in nitrobenzene are only the ones present inside the benzene structure, hence the pi bonds in the group NO2 is ignored and Huckels rule is followed similar to that of Benzene structure itself.

Any substituent attached to Benzene that does not distrupt its delocalization of electrons inside the ring can be termed aromatic.