I don't think any of the sentences is correct
a. Though he is poor,
so he is large hearted. (if a person is poor then he/she must be large-hearted?. No no subtle rule regarding that)
b. Since body and mind
goes together, illness of one inevitably affects the other. (goes is singular and noun here is not)
c. Everybody admires
them because
he is honest. (noun singularity and plurality confusion)
d. She will not go out if it
rain. (it has to be "rains". The noun rain is usually uncountable and would take a singular verb. However in weather forecasts, when we are talking about it raining at different times in different places, we sometimes talk about heavy rains, which is plural.)
e. He was rewarded
as he was truthful. (The sentence depicts 2 things, he was rewarded because he was truthful. he was rewarded as soon as he was truthful (which means he was rewarded after he was truthful, sentence confusion)
Can you share the source?