I am having some trouble to put things together into one picture. Can a modulated carrier wave be a square wave? If yes then How? Because at resonant frequency at the transmitter, the current oscillates between L and C. It takes time to charge and discharge and not abrupt. So how a square carrier wave is achieved? If No then can I ever generate a (say) FSK modulated square carrier wave? Also will there be any advantage/disadvantage (in terms of say transmission loss, energy per bit, suppression of higher order harmonics) of using a square wave over sine wave?
When sending data over a cable or as light pulses, modulated square waves are common since it's easier to produce square waves than sine waves. When sending radio waves over the air, however, there are generally very restrictive rules about transmitted spectral content; because of those rules it is generally desirable to avoid using much bandwidth than would be necessary to send information.
If there were no other radio devices in use and one was allowed to transmit whatever and however one wanted, it's possible that sending a modulated square wave might allow better performance than would be achievable with the same level of power using a comparable amount of circuitry; many kinds or resonant structures inherently behave like a comb filters, which will very nicely capture all the harmonics of a square wave just as effectively as the fundamental.
Nowadays, however, one isn't allowed to transmit whatever and however one pleases, but is instead required to confine transmissions to particular frequency bands. Amplitude-modulating 300-3300Hz audio content on a 1MHz square wave would generate spectral content from 996,700Hz to 999,700Hz; 1,000,300Hz to 1,003,300Hz; 2,996,700Hz to 2,999,700Hz; 3,000,300Hz to 3,003,300Hz; 4,996,700Hz to 4,999,700Hz; 5,000,300Hz to 5,003,300Hz; etc. The higher-frequency spectral content might help slightly with one's ability to receive the signal, but the potential improvement in receiver performance would be nowhere near sufficient to justify the huge increase in spectrum usage.
Yes you could modulate a square wave and it could act as a carrier. It is not a given that transmitters or antennae need to be resonant, or - if they are - highly tuned, though it is usually desirable that they are.
An untuned square wave transmitter will be received at its fundamental frequency F, and at every odd harmonic, 3F, 5F, 7F etc with decreasing power at higher harmonics. This is usually undesirable as its principal result is to interfere with other users of the spectrum.
So in practice a sinusoidal waveform delivered to the antenna is highly desirable.
However it's not unusual for an RF power amplifier to work in "Class C" where the power stage itself is highly non-linear (with a spectrum similar to a square wave) It delivers power into tuned circuits which pick out the fundamental and reject the harmonics.
In the amateur bands where home-built transmitters are permitted, which may not be perfectly tuned, this is sometimes mitigated by designating another band at the 3rd harmonic as another amateur band! Hence in the "2 metre band" (144MHz approx) the strongest (3rd) harmonic will lie in the 70cm band (around 432 MHz).