Would it be correct to say that ozone is an element? My level of chemistry knowledge is restricted to that which is given at high school level. 

I was tought that an element is a substance that consists of only one type of atoms, meaning the number of protons is the same for every atom. Is this definition sound? 

In case it is sound:

Since ozone gas only consists of oxygen atoms, would it be correct to say that ozone gas, as an entity, is an element? 

Would it be more correct to say that ozone is an allotrope of the element oxygen?

Why is it called the table of elements, when it seems that we are actually listing the atoms and not the elements themselves?
It would be more correct to describe ozone as an allotrope of oxygen. Dioxygen (about 1 in five of the molecules making up air) is another allotrope.

The periodic table describes the atoms that make up elements as this is the most fundamental property. Many other elements, though, exist in multiple allotropes. Carbon, for example, has several: graphite, diamond, buckminsterfullerene and quite a few others. 

All allotropes consist of only one type of atom but the bulk physical properties depend on how the atoms are joined up (flat plates of carbon hexagons with multiple bonding in graphite, a three-dimensional network of singly-bonded tetrahedral carbons in diamond). 

We don't normally add too much about allotropes in a periodic table as it would make the table a little big, although some include limited information about how many there are. And, usually, only one is thermodynamically stable under normal conditions (so the physical properties will be about that one).