When something reaches the speed of light, then time is said to stop, or at least is perceived to have stopped. The reason is that at that speed, the object become infinitely heavy and requires infinite energy to move. My question is, why does light itself (electromagnetic wave) continue to move, or move at all, if it is travelling at the speed of light, and should be frozen in time? My own best guess is because light doesn't have mass, and so does is not affected by $E=mc^2$. But then why aren't other energy-only waves able to travel at light-speed? As for the implications, would it be possible to travel at the speed of light if an object could be converted into light and then back somehow?
Lots of questions! I'll try to get each of them:
the object become infinitely heavy and requires infinite energy to move
Light (photons) have no rest mass - ten, a hundred or a million times nothing is still nothing, so the (non-existent) mass isn't relevant - as you say further down. (Light does, counter-intuitively, have momentum, but that's another story).
why does light itself [electromagnetic wave] continue to move, or move at all, if it is travelling at the speed of light, and should be frozen in time?
This depends on your perspective - if you're "riding on the photon" you'd find that it doesn't travel - no time passes - and the point where it "starts" and "finishes" are effectively the same. In technical terms, "the spacetime interval for any light-like path is always zero". From the perspective of us, sitting on Earth, and looking up at the Sun, yes, the light takes ~8 minutes to get here.
But then why aren't other energy-only waves able to travel at light-speed?
They are...probably. Both theory and current experimental evidence show that gravitational waves, for example, propagate at c. ("Speed of light" is a slightly misleading term in that regard.)
As for the implications, would it be possible to travel at the speed of light if an object could be converted into light and then back somehow?
Yes, absolutely. If I was able to read every last detail about what made up "you" and then email it down a laser beam to the Andromeda galaxy, where it's then recreated by a magic box, then you'd experience zero time between your data leaving Earth and arriving at Andromeda, 2.5 million lightyears away. There's good reasons why this is impractical/impossible in practice, but if you were to write the idea into a SF story it'd be considered accurate, relativity wise.