Why is a laser beam monochromatic and coherent? 
Why is laser beam monochromatic and coherent?

My reasoning: Laser is monochromatic because there is equal energy difference between metastable and ground states.
Laser is coherent because there is no time and path differences between wave-fronts of any two directions (beams).
I am not sure if my reasoning is correct. Any explanation will be greatly appreciated.
 A: This is actually an excellent question because I feel that the actual operation of lasers is usually dumbed down to a simple narrative of "in-step photons" that come out of a hole. Even by people in the field. This view, whilst technically incomplete, does give you the right answers so that you can "work" with lasers. But it does not satisfactorily address the conceptual fallacies.
Monochromatic
Ideally, sure, the frequency of the photon is exactly equal to the energy difference between the two levels. So it should be 100 % monochromatic.
Not quite.

*

*Even if the states used in lasers are metastable, as you correctly point out, they still have a finite lifetime, albeit very long. This will give the atomic transition a natural linewidth which is the "physical" limit to how good your laser could possibly be.


*Another shift may come from collisions or Doppler shift of the atoms in the gain medium (if this is not a solid, like in a dye laser).


*Finally, there may be several cavity modes that are resonant with the light emitted by the atoms. This introduces another broadening, cavity-dependent (related to its FSR).
All these effects lead to a net broadening, which is usually quoted in the data sheet.
Coherence
The usual narrative is that "laser light is coherent because the photons are emitted in-step/in-phase". No, this is not true. What the in-phase statement does result in, however, is the amplification effect of a laser (incidentally, the 'A' in lAser) just by virtue of constructive interference.
To ask why is the laser coherent is equivalent to asking why are the photons emitted "in-step" / "in-phase"?
Let's first review some basic definitions.
Definitions
Temporal coherence is the ability of light to maintain a constant phase at one point in space at two different times. This essentially follows from the light being monochromatic, and in fact the coherence time $\tau_{\mathrm{c}}$ is related to the frequency width of the laser light $\tau_{\mathrm{c}} \propto 1/\Delta f$.
Spatial coherence: is the ability of light to maintain a constant phase between any two points selected at random over time. This is quantified by the coherence length which is $c\times \tau_{\mathrm{c}}$ so it's not independent of temporal coherence.
"In-phase" photons $\Rightarrow$ amplification, but "in-phase" photons $\not\Rightarrow$ coherence.
The usual narrative of "a laser emits in-phase" photons can only really explain the phenomenon of amplification. Which is none other constructive interference. You line up two waves so trough-to-trough and peak-to-peak and you get a wave twice as big. So not only it does not address why laser radiation is coherent in the first place, but amplification is actually a consequence of the light being coherent.
The fallacy of "in-phase light $\Rightarrow$ coherence" can be readily exposed by noting that even incoherent light going through glass gets absorbed and re-emitted 'in phase' so that you see it as transparent and not hazy/translucent.
Fixing temporal coherence
How do I introduce a bit of temporal order? So that I know what atoms are doing when?
Use metastable states with long lifetimes, that don't spontaneously decay whenever they like (in like ns or shorter timescales). Hence, when one of the pumped photons comes in inducing stimulated emission, you are 'sure' that's when the other photon is going to get emitted. Ordinary light is not temporally coherent because of the short lifetimes of the excited states and hence the increased (short-term) randomness in the emissions.
But ordinary light is also spatially incoherent light because it does not come from a single point. Different atoms are emitting light all along the tungsten filament (for a lightbulb). Each starting point leads to a phase delay etc. A diffraction grating does not have this problem because the slits are separated by a constant $d$. But think about superimposing $N$ gratings each with random $d$'s and you quickly get a mess. But even here note how the "peaks appearing only at certain positions" is related to the constructive interference ('amplification') due to the "in-phase" (read: spatial coherence) of the light coming out of each slit.
Solving spatial coherence
To solve the spatial coherence problem, you "just" have to force the laser light to behave "as if" it is coming from a single point. This is achieved by having a cavity, with its standing waves - that will not leak and that will hence lead to the amplification that you then see at the output. The image below gives the right idea, but you should think of the cavity mirrors as being curved, so as to support a spherical wavefront:
$\hskip2in$
Making coherent light from incoherent light
To sum up all of the above, you can start with an incoherently emitting lightbuld and mae into a coherent light source by:

*

*adding a spatial filter (pinhole - 'fixing' the spatial coherence), and

*adding a frequency filter ('fixing' the temporal coherence),

as shown below.

A: 
Why is a laser beam monochromatic?

Well, it's not. At least not in the general case. There are lasers which are specifically built to be non-monochromatic: Mode-locked lasers. (https://en.wikipedia.org/wiki/Laser#Mode-locking)
These lasers include some means to turn the light that cycles around the laser cavity into a single extremely bright pulse. The Fourier transformation of such a single pulse reflecting back and forth in a fixed cavity yields a wide comb of wavelengths. In the extreme case, you can build lasers that emit white light in the form of thousands of very thin spikes in the spectrum that work together to form a broad spectrum.

As to coherence: After any significant time of operation of a laser, a single mode usually wins out. The light that comes out of the laser has been reflected and amplified within the laser cavity millions of times, and because of these many, many, many amplifications, any differences in gain are extremely significant. So, if you have two incoherent photons start amplification within the laser cavity, one may experience just a tad less amplification or more attenuation. However minuscule the difference in effective amplification of the two waves, let's call the difference factor $\alpha$, after a million round trips through the cavity, the difference factor will be $\alpha^{1000000}$. I.e. to have a laser that outputs two noncoherent waves, the effective amplification factor must be exactly equal. If there is any difference in the amplification factors, any difference at all, only one of the two waves will win out, producing coherent output from the laser.
You need to add special elements into the laser cavity to force a laser to produce more than one wave. This is what a mode locking laser does. Effectively, it turns the entire file of coherent frequencies into a single joined mode. Without such a coupling, the laser will just choose one wave for you.
