In a well-written English sentence, the relative clause usually refers to the last noun or noun phrase before the comma which sets off the relative clause:
The battle was fought in the field of PlassyANTECEDENT, an old city of Murshidabad.RELATIVE CLAUSE.
(In the following, we will replace the verb exist with the verb sit. It is is not idiomatic in this context to use the verb exist; instead, we would use the copular be or another verb like sit.)
Your second sentence omits the relative pronoun "which":
The battle was fought in the field of PlassyANTECEDENT, [which]RELATIVE PRONOUN sat beside the river Ganga.RELATIVE CLAUSE
As you noticed, it is unclear, even with the addition of the relative pronoun, whether the relative clause refers to the proper noun "Plassy", or to the preposition phrase "field of Plassy". This is the fault of the writer, and not of the poor reader! To clear up the confusion, we could use "Pied-piping" to repeat the noun that heads the preposition phrase:
The battle was fought in the field of PlassyANTECEDENT, [which field]RELATIVE PRONOUN sat beside the river Ganga.RELATIVE CLAUSE
This, however, is archaic English. A better version of the sentence will make the antecedent quite clear:
The battle was fought in Plassy on a fieldANTECEDENT [which]RELATIVE PRONOUN sat beside the river Ganga.RELATIVE CLAUSE
Even better, the writer can dispense with the relative clause entirely:
The battle was fought beside the river Ganga in the field of Plassy.
Your other examples present other difficulties, but they are not as obvious as the first.
My father used to visit the children of his familyANTECEDENT, which is very big.RELATIVE CLAUSE
Here, the antecedent is clear: the last noun before the comma is family, and any English speaking reader will understand that it is the family which is big.
Your next example is ungrammatical in English: it omits the necessary relative pronoun which. With the pronoun in place, we have:
He talks about the subject of the paintingANTECEDENT, which hung on the wall.RELATIVE CLAUSE
Here again, the context makes the antecedent clear to an English-speaking reader: a subject cannot hang upon a wall!
He talks about the subject of the paintingANTECEDENT, which is taken from Bible.RELATIVE CLAUSE
Here, although most English-speaking readers know that the Christian scriptures do not include any paintings, some confusion may arise. To resolve the problem, we can use the Saxon genitive instead of the preposition of to indicate possession:
He talks about the painting's subjectANTECEDENT, which is taken from Bible.RELATIVE CLAUSE