First of all I think there is (can be) slight misconception about the meaning of 罹る even among natives.
罹る means to become a state of serious illness, or falling in serious sickness (thus the answer for question 2. would be technically 'punctual verb' in this aspect) and it rather focuses on the action than the state of illness compared to its synonymous verb 患う {わずらう}.
患う can be translated fairly well as 'to suffer from sickness'. 患う can connotes more of pain/torment/hardship of being sick than 罹る.
I also recommend you to see also 罹患する
Let's look at the sentence on the above basis.
長男を失った母親として、ガンに( )娘の母の気持ちは共感するにかたくない。
In this context, I assume you want to stress the hardship aspect of the daughter who
1 罹る
2 罹った
3 罹っている
4 罹っていた
a cancer.
If so, I'd like to alter 'ガンに罹る' to 'ガンを患う' here.
Also (this is not relating to your question but...) 共感する is technically making sense but is mainly used when sharing positive feelings, so I would like to change it to [汲]{く}む (in the link definition 3)
長男を失った母親として、癌を( )娘の母の気持ちは汲むにかたくない。
1 患う
2 患った
3 患っている
4 患っていた
This might be able to convey your intention better.
And the the answer for question
- Which of the following choice(s) would be appropriate, and what would the meanings be?
would be grammatically all the choices but the context would narrow the choices onto 2 罹った and 3 罹っている provided that we could put the choice of word aside. (sic basis)
the reasons
1 罹る is not really appropriate.
Here, 罹る is present tense thus the daughter doesn't have a cancer yet. The daughter is not suffering yet and she possibly recuperates from cancer.
It's a bit unreasonable that the deceased son's mother shares the feeling with the yet healthy daughter's mother.
2 罹った and 罹っている are appropriate.
Daughter must be suffering from cancer, which is compatible with the context.
4 罹っていた can be appropriate
only if the daughter is dead.
We cant tell if she's dead or well now from the sentence so I don't think this is good.
Lastly, I'd like to answer the same question on basis of my corrected sentence.
All of the choices would be appropriate, because regardless of the tense of 患う the two mothers have it a common that their children suffer(ed) from cancer.