1) Does the cloud base refer to a single cloud hence it is not significantly used
among pilots ? compared to ceiling which is a group when it covers more than
half of the sky
Clouds, in a METAR have their bases defined as height above ground level. That's for all clouds regardless of opacity. A ceiling is defined as any cloud base obscuring more than a certain amount of the sky, either given as BKN or OVC. Therefore, a ceiling is simply the base of cloud for which BKN or OVC is given.
[2) Is airline pilots really interested in cloud base or ceiling as they always fly IFR , even though they can look out.
Even in the IFR world all landings (except those rare, CATIIIc auto lands) require visual reference to the ground in order to land. We are mostly concerned about the ceiling (cloud base for BKN and OVC) being low, but any cloud that obscures our vision below our minimum IFR altitude is also cause for concern.
3) Only VFR pilots and Flying schools worry about these since for VFR ceiling checks are mandatory?
Remember, VFR means you cannot into a cloud or near a cloud within a specified distance. Therefore, flight schools have to weigh the probability that you would break those rules during takeoff, cruise, or landing. There may also be minimum weather conditions for a school in order to obtain insurance which may be much more restrictive than your regulatory body requires. These may, or may not, change with your level of experience. It's the same in the IFR world (about concerning ourselves with the lowest ceiling) but our minimums are much lower.
Hope that helps!