Why does Jefferson believe that answering questions about one's faith would compromise the "right of independent opinion?' (In reference to Letter to Dr. Benjamin Rush.)
I don't even understand what he means by "the right of independent opinion."
What he means by "the right of independent opinion" is the right to have religious opinions of your own that are not necessarily like those of the average person in your society. Jefferson is saying that, if people start answering questions about their faith, no one will be able to have opinions that are unlike those of the mainstream.
The reasoning here is that answering questions about your faith exposes your beliefs to public scrutiny. If they are not in line with the ideas of the majority, you can get ostracized. If some people answer questions about their faith, then everyone must because those who don't will be suspected of having "bad" opinions.
Look at it this way: imagine you and I are running against one another for some office. You have conventional religious beliefs and I don't. You answer questions about your belief. If I don't, people will suspect that I have "bad" beliefs. If I do, they'll know that I have those beliefs. And yet our religious beliefs are supposed to be a matter of our private opinion and faith.
This is why Jefferson does not think people should answer questions about religious faith.