Does this indicate ownership by my mother?
This clearly indicates that the beneficiary enforced the will and taken possession, however since it is mandatory to obtain probate, you may have to it in order to have a clear and marketable title to the property.
b. Can my parents put the above property on rent with the non-probated
will and the details provided on point 'a' above? Can a lease/ agreement be drawn based on the above?
They can rent or lease out the property on the basis of the will in their favor.
c. If yes, can my mother and father sign the lease agreement as lessors?
Since your mother is the owner or beneficiary of the will, it is her property hence she only can enter into the lease agreement.
d. Can any legal complications arise if my parents decide to give the property on rent?
In the given circumstances, there should not be any problem.
e. My parents have occupied the property on and off for the last 20 years, unopposed by anyone before my grandfather died in 1998 and even after that.
Law of adverse possession will not operate because for partition there is no such restriction, hence let your mother get the will probated through court competent.
f. My father also built an extra room which is much bigger in size in the property. My father also has written acknowledgement of this expense.
That is his problem because he is not the owner, hence his expenses cannot entitle him a legal ownership.