Have we proven that higher dimensions exist? Have we proven higher dimensions exist?
 A: To start with in physics we do not prove basic concepts. Theoretical models are mathematical constructs which, together with the postulates of the theory, fit data and predict new behaviors. Physics theories can only be falsified, or validated by experimental results. Not prove, because they are not pure mathematics with QED attached in proving theorems . To be relevant to measurements and observations they depend on postulates, extra axioms that allow the connection of mathematical solutions to measurements and observations.
The current theories which fulfill these conditions use the spacetime four dimensions, and their predictions are continuously validated, and never falsified. This gives the confidence that the model of four dimensional spacetime fits the data.
In research string theories offer many more dimensions than the four ones, but they are still at the research level and no standard model exists. No new predictions to be validated or lead to falsifications have been proposed. If/when research reaches this point and the data validate string predictions then we can talk of the existence of higher dimensions than four in our model of physical reality.
A: No.
There are various ways we could prove that extra dimensions exist:


*

*observe Kaluza Klein excitations

*observe microscopic black holes

*observe deviations from the inverse square law of gravity at very short scales

*find proof for string theory
Option 1 would be the most likely option. If there are compact extra dimensions then particle wavefunctions can wrap round them and this would produce a characteristic spectrum that could in principle be observed in accelerators.
Options 2 and 3 would only work if the length scale of the extra dimensions is relatively large. If their length scale is around the Planck scale we'd never be able to build an accelerator powerful enough to make the observations.
Option 4 would be if compelling evidence for string theory was found. Since string theory requires extra dimensions this would be an indirect proof that extra dimensions exist. Finding evidence for supersymmetry would be a good start, but so far no such evidence has emerged.
So far no evidence has emerged for any of the options above.
A: I think dimension means the number of independent coordinates required for describing or specifying the position and configuration  in space of a dynamical system. As example when we consider the Minkowski's four-dimension world we consider the position coordinate is homogeneous to time coordinate. So an event is represented by $(x,y,z,\tau) $ where $\tau=ict$. And in this space $s^2=x^2+y^2+z^2+{\tau}^2=x^2+y^2+z^2-c^2t^2$., which remain invariant under Lorentz transformation. 
As far as you say about the graphical representation of these fore coordinates, we plot the time in imaginary axes.
So, yes. Higher dimension exist.
