How strict are the boundaries that divide dimensions? Is a single-layer sheet of graphene 2D or 3D? I would like to know if there is any theory that describes a set of rules that define the boundaries of dimensions.
For example, does a single layer sheet made of graphene considered a two or a three dimensional object?
Or does any object consisted of at least one atom considered by default a 3D object?
How about the elementary particles, do they qualify as 2D objects?
Since a 3D object can be projected to and be observed in a higher dimension, do we have any example of 2D objects projected to the third dimension?
 A: 
For example, does a single layer sheet made of graphene considered a two or a three dimensional object?

Three dimensional , as all atoms and complex systems

Or does any object consisted of at least one atom considered by default a 3D object?

Complex systems, including protons in these, are three dimensional. In addition when the quantum mechanical uncertainty relation is taken into account, the dimensional location becomes fuzzy at the microscopic level.

How about the elementary particles, do they qualify as 2D objects?

Elementary particles are point particles, i.e. zero dimensional.

Since a 3D object can be projected to and be observed in a higher dimension, do we have any example of 2D objects projected to the third dimension?

I know of none such, at the moment higher spatial dimensions than three are not within standard physics models. In general complex systems of elementary particles are three dimensional.
A: As anna already answered, rigorously, any object is either zero- or three-dimensional.
That said, it can be very useful to approximate a 3D object by a 2D or 1D one. Graphene in particular can often be well approximated as a 2D sheet.
Being an approximation, there's no strict rule about when it can be used: as long as it provides a description close enough (for your purposes) to reality, it's a good model.
A: As others have pointed out, even a single atom exists in three dimensions. However, there's an alternative, and mathematically rigorous, sense in which you should regard a sheet of graphene (or two sheets of stacked graphene, or four sheets of stacked graphene) as 2D.
In condensed matter, one is often interested in the properties of a material in the thermodynamic limit, i.e. as $N\rightarrow \infty$, where $N$ is the number of atoms in the system. There are many ways of taking the thermodynamic limit. If we consider a cube of material with dimensions $L_x\times L_y \times L_z$, we could take $L_x\rightarrow\infty$ while leaving $L_y$ and $L_z$ unchanged, or we could take $L_x,L_y\rightarrow\infty$ while leaving $L_z$ unchanged, or we could take all three to infinity. We call the first case "one dimensional" no matter how big $L_y$ and $L_z$ were initially, we call the second case "two dimensional" no matter how big $L_z$ was initially, and we call the third case "three dimensional." In other words, a one dimensional system in condensed matter is a system where one direction is infinite, and the other two directions are finite. In a sense it doesn't matter how large the other two directions are, since they are necessarily negligible compared to infinity!
With this definition of dimension, you can prove all sorts of interesting theorems. You can show a continuous symmetry cannot be broken in dimensions $\leq 2$. You can classify topological insulators according to their dimension. And on and on. All of these theorems are proven using the notion of "dimensional" that I defined above.
