What is the physical significance of beta and alpha in BJT?
I never (need to) use \$\alpha\$ f , I always use \$\beta\$ (same as \$h_{FE}\$) for all configurations: Common emitter, common collector, common base.
If you make a schematic drawing and indicate the currents \$I_b, I_c\$, and \$I_e\$ and using \$\beta = \frac{I_c}{I_b}\$ then it is easy to write down \$\alpha = \frac{I_c}{I_e}\$ and then re-write that to express that using beta.
My point: using \$\alpha\$ for a common base is a choice. You can use it if you want and it makes your life easier. However I'd rather use \$\beta\$ as \$\beta\$ (or \$h_{FE}\$) can be found in the datasheet of a transistor. \$\alpha\$ is not in the datasheet but if I know \$\beta\$ then it is easy to know what \$\alpha\$ is. So I just don't need alfa.

Why are we calculating these current gains?

\$\beta\$ Is a current gain and a property of the transistor, as long as the transistor works in active mode, \$\beta = \frac{I_c}{I_b}\$ is true.
I see \$\alpha = \frac{I_c}{I_e}\$ more as a &quot;helper&quot; expression since it is mostly only useful in the common base configuration as in the common base the emitter is the input and the collector is the output. So really \$\alpha\$ is just the transfer function of the common base.

And Why is there a range of beta for a particular transistor? What does minimum and maximum values of beta mean?

Because it is almost impossible to manufacture transistors with a precise value for \$\beta\$. \$\beta\$ depends on the width of the base and doping levels which are hard to control. Some transistors have a very large range for \$\beta\$ like between 30 and 300.
In the factory all fabricated transistors are measured and the ones that do not have a 30 &lt; \$\beta\$ &lt; 300 are discarded.
You can buy transistors with a narrower range, for example 100 &lt; \$\beta\$ &lt; 200, then those transistors are &quot;binned&quot; meaning, they're measured and then marked depending on what their \$\beta\$ is.
To beginners this large range of \$\beta\$ often sounds like a huge problem. In reality it is not, we can design our circuits such that they can work with a \$\beta\$ that is for example at least 50. Then we just need to use transistors that have a \$\beta\$ that is always larger than 50, like a transistor that is specified to have: 100 &lt; \$\beta\$ &lt; 300. Yes, that 100 is 2x larger than the 50 we wanted but that's actually good meaning we get some &quot;design margin&quot;.

How much does these alpha and beta affects our Early Effect?

The Early effect makes \$\beta\$ appear smaller as \$V_{CE}\$ increases and that is due to base with modulation. There is no easy to understand direct relation so as a beginner, don't worry about this too much.