Will separating a stage(first) immediately after its burnout cause damage to the upcoming stage (2nd stage).?
While there are some exceptions such as the Soyuz 2-1v which is specifically designed to start the 3rd stage while the 2nd is still burning, there is almost always a delay between one stage burning out and one stage igniting. Stages do not burn out all at once; there is a residual amount of thrust that tapers off over a few seconds. This is usually accounted for by the delay between stages, but this was the exact mode of failure for a Falcon I launch in 2008. It seems they miscalculated the delay needed between MECO and staging.
"If there had been one extra second, this wouldn't have happened"
Here is a very dramatic video of the Titan II performing a similar type of staging (Thanks @Tristan)
Also,do the first stage separation and second stage ignition happen at the
same instant ??
No, there are usually at least a few seconds between stage separation and ignition. This timeline of the Falcon Heavy maiden flight shows an 8 second delay (from T+0:03:07 to T+0:03:15) between the two events. This is especially critical for reusable rockets such as the Falcons because the first stage must not be damaged.