Can a droplet of water bounce back when it hits a water surface?

Yes, as you can see in this video. As you can see, the droplet will hit the surface, partially coalesce (merge) with the bulk, re-emerge as a smaller droplet, bounce 1-3 times, partially coalesce again, re-emerge again as an even smaller droplet and so on. This process is known as coalescence cascade. You can find another video here. Eventually, the coalescence will be total and the droplet won't re-emerge anymore.

This if the surface is still: but if it is vibrating, another really interesting phenomenon can be observed (video). As you can see, in this case the droplets don't coalesce with the bulk, and will stay intact while "jiggling" on the surface until the vibration is switched off.

if you watch extreme slow motion film of a single water droplet falling into other water, usually a completely flat horizontal surface, you see many circular ripples being created, most of the droplet being absorbed by the body of water, but a tiny ( about 5% of original drop ) droplet being formed, and returning back, vertically upwards.

How much of this is the falling droplet I am unsure of, it may me mostly from the target pool.

I suggest a further experiment, where the falling droplet contains an intense dark dye, falling into completely pure water, filming with a high frequency stroboscope or fluorescent exciter, to further determine this.