After having used FF a few times, I realized there are features other than failure to download/install that IE fails to do while FF does(foremost of which is adult sites video streaming).
I still do not know how it came about. I was suspecting spywares but several anti-viral programs scans including several scans by adwcleaner failed to pick up any malware.
So...I decided to uninstall IE11.
After uninstallation, I was surprised that I was reverted to IE8(I was expecting IE9 or 10). With IE8, update check showed recommended cumulative updates for IE8 and several optional updates: including IE9, IE10 and IE11. I selectively OK'd IE11 and ignored the others.
With IE11 reinstalled, several updates for IE11 were recommended, which I installed.
And now, I am back to normal. Everything FF does, IE11 also does...including installing downloaded programs/software.