It's not ihr es, but ihrs / ihres (theirs).
I think, today you would rather say something like:
Sie dachten sich ihren Teil dabei.
They had their own thoughts on the matter.
But as you see: der Teil is nothing really specific. It just stands for what comes to their mind.
A sentence like
Ich denke mir meinen Teil und schweige.
I keep my thoughts ...
The only reason for a noun phrase to be accusative is that a verb or a preposition requires the accusative. The case required by a verb or preposition is fixed for each syntactical role. It doesn't depend on the rest of the sentence.
Thus, the accusative in sentence 1 and 2 has nothing to do with the pronouns, but is required by the verbs jemanden/etw. mögen ...
It does have to do with the pronoun in your examples, but you can't generalize that.
The important question is: in your "dass"-clauses, who or what is the subject, and who or what is the object? The subject is in nominative, the object is in a different case, most of the time accusative or dative.
For your examples that means:
Meine Mutter findet, ...
No. Verbs and prepositions govern the case. The verb sein demands the nominative:
Meine Mutter findet, dass ihre Ärztin gut ist
On the other hand etw. haben or etw./jmdn. mögen demands the accusative case.