Is the Physics for extremely large and extremely small particles similar in any aspect? Do the equations we use to describe natural phenomenon on a very large scale, work as well for describing natural phenomenon on an extremely small scale as well?
Is there any correlation at all which would suggest that any natural phenomenon correlates when we take extremely large and extremely small quantities into consideration.
 A: 
Is the Physics for extremely large and extremely small particles similar in any aspect?

It is similar that it uses similar differential equations at all scales, which obey the same conservation laws at all scales ( exception conservation of energy and general relativity at large scales). 

Do the equations we use to describe natural phenomenon on a very large scale, work as well for describing natural phenomenon on an extremely small scale as well?

No; there are usually different ( but similar) differential equations for each scale, the underlying frame being quantum mechanics for particle dimensions;, all other frameworks emerge smoothly from that sequentially . Gravity on the relevant scale, human scale, also has different equations that blend smoothly with the ones of General Relativity for humongous masses.
The exceptions are chaos and complexity . Fractals can appear in any dimensions depending on the boundary conditions , as well as turbulance as mentioned in the comments when the framework  allows fluidity.
A: No, on the very small scale, quantum mechanics applies It differs a lot from the observation we make in the macro-world, but the differences in quantum mechanical measurements are (usually) so small when applied to macroscopic objects that they can be ignored.
A: It is believed that momentum, energy and angular momentum are conserved on all scales.
