Why is the mol a fundamental physical quantity? I am starting to study physics in detail and as I read about physical quantities, I was puzzled why mol (amount of substance) is taken as a physical quantity.
A physical quantity is any quantity which we can measure and has a unit associated with it. But a mol represents the amount of substance by telling the number of particles (atoms, molecules, ions, etc.) present. So it is a pure number and numbers are dimensionless. So mol should not be considered a physical quantity.
Also, fundamental physical quantities should be independent of each other. I am wondering whether mass and mol are independent. This is so as they surely affect each other as we can evidently see while calculating the number of moles and using the mass of that sample for calculation.
So how is the mol a fundamental physical quantity and independent of mass?
 A: Is it a fundamental number in nature? It's (currently) a number resulting from atomic structure (fundamentally defined by the masses of quarks, Planck's constant and the way quantum mechanics works) and our definition of the gram, which is based on the international Kg prototype. Avogadro's constant is currently defined by experiment, and therefore has no absolute "right" number, just an agreed working definition.
This is a messy way to define things though, and there are many arguing that the Kg should be defined in terms of a particular element and Avogadro's number, which would put it on a more "fundamental" level in my book.  (See https://en.wikipedia.org/wiki/Kilogram#Avogadro_project )
This would mean "fixing" Avogadro's constant by simply picking a number, then defining the Kg in terms of this, in the same way the second was "fixed" in terms of the so-many-oscillations of a particular frequency of light, rather than being a 60th of a 60th of a 24th of one rotation of Earth (a messy, variable number). 
A: The mole definitely isn't a fundamental physical quantity. It's just a shorthand for Avogadro's number, to make really big numbers more tractable.  It's purely there for convenience, there's nothing fundamentally physically significant about it.
A: Mols are a units of quantity. Technically, you can have a two cars, or a mole of cars, two forks, or a mole of forks, two baby rabbits, or a mole of baby rabbits. But since one mole is such a large number, it is only really useful for things that you have lot of, like molecules. In that case, though, it is very useful, since saying one mole is lot faster than enunciating 602 sextillion, or $6.022140857\times10^{23}$.
It is very important to know how many molecules of a particular type there is (for instance) in a beaker. If you have two highly reactive molecules in a beaker, it's probably not too dangerous: these two molecules will only destroy two of the floor's molecules were the beaker dropped. However, if you have a mole of these dangerous molecules, the floor might start to complain.
Mass and mole are completely different things: a mole of cars will weigh more than a mole of H2 molecules.
A: True, a mole is a measure of quantity, i.e. it's dimensionless. But that doesn't prevent us from treating it as a physical quantity!
The fact is, units can be treated the same way as the numbers (or symbols) that they apply to - multiplied, divided, reduced, replaced with equivalent expressions.
The special feature of dimensionless units is that they can turn from number to unit and back at will. E.g. when transposing units, you can always replace the $k$ prefix with $10^3$, $M$ with $10^6$ etc and vice versa.
See:
$$
20\times10^{23} \approx 3,3\,mol\\
5\,mol \cdot 5\,g/mol = 25\,g\\
N_A=6,022\times10^{23}\,mol^{-1}=1\\
R=kN_A=1,380\times10^{-23}\,J/K \cdot 6,022\times10^{23}\,mol^{-1}=8,314\,\frac{J}{mol\cdot K}
$$
"$mol$" is effectively just a multiplier, so $R$ is actually $k$ in different units!
Since $mol$ is dimensionless, you can legitimately (in the mathematical sense) introduce it anywhere in any power. But that likely won't make physical sense since there's no such thing as a "square number of atoms" - so once introduced in an appropriate place, it should rather be treated as a dimension from that moment on to keep everyone's sanity intact :)

So, a mole is often considered a physical quantity because it's convenient to treat is as such - this results in more comprehensible numbers in the numerical part of expression when dealing with practical amounts of substances. Besides, it points out that the number that "$mol$" is used with is not just any number but a number of atoms - a plain number wouldn't carry this meaning.
One other dimensionless entity commonly treated as a physical quantity is the decibel.
