Can light continue without a light source?
Yes, it would continue. But not forever, for two reasons. One is that no mirror is perfect, so a bit of light is lost at each bounce. The other is that no beam of light is perfectly parallel ("collimated"), so that the light spreads out over time, and light eventually falls outside the mirror.
Edits after comments
Spherical mirrors will help, but they will not eliminate the problem at hand, which is diffraction. A perfectly collimated beam is not possible, but as a corollary to that it is also not possible to produce a beam with a finite cross section. Some portion of the beam always falls outside of the next mirror.
Using beams that are approximately Gaussian (perfect Gaussian beams are impossible) and spherical mirrors the amount of energy that misses the next mirror is small, but not zero.
The OP's question #3 is new. I don't quite know what you mean by "itself as the source". If you can somehow get the light going, it will continue for a while after you turn off the source. The decay, as others have pointed out is (approximately) exponential as roughly the same fraction is lost on each bounce. It's approximately exponential because the beam shape will change as bits are diffracted away, so the fraction lost changes slightly from bounce to bounce.
How long the light will persist depends entirely on the quality of the set up. The surface quality of the mirrors, their surface figure, the atmosphere, the materials used, the rigidity of the mounts ... It's probably possible to estimate the longest possible persistence time taking into account effects that others have mentioned in other answers: scattering, heating, momentum transfer, ... I don't know what the "theoretical" maximum would be, or the practical limit. A quick search on Fabry-Perot interferometers finds finesse values in the $10^6$ ball park, which would imply a persistence time for a 30 cm cavity of about 1 ms, but that's a very rough estimate.
would the laser light line continue to exist between the 2 mirrors if the source of the laser stopped.
As DJohnM commented,
light travel to the moon and back, even though the laser source is off for most of the round trip.
This happens because any light source emits photons. They are indivisible packages of energy and they are traveling as long as they not get absorbed by an obstacle or - more precise - by a subatomic particle.
Would it just continue existing between the mirrors using itself as the source or is this not possible for light?
As told above, once emitted, the photon is on it own and don’t anymore care about the source.
But where is another point about the mirrors. What garyp told you in his answer is about the technical imperfection. Beside this, any reflection process of photons is accompanied at least with the transfer of a momentum to the mirror. Every photon pushes the mirror a little bit and any photon - due to energy conservation - leaves the mirror with a lower frequency. So the light dies a infrared death and the mirror gains velocity, or the temperature raises.
We recieve light from far away stars even they are dead when the signal reached to earth. So, the source is not there when we actualy see it (capture the light with space telescopes).