Let's leave "intelligent" aside and focus on the "rational" necessary condition for being "intelligent and rational." Also, let's dig down past the label "conservative Christian" (or "conservative Catholic," as Chris actually said) to some of the beliefs that constitute conservative Christianity and conservative Catholicism. For example, in the American context, a conservative Christian who isn't Catholic is probably some variety of creationist, and quite likely a young-earth creationist. Finding out that a person is a YEC would reduce my probability estimate that that person is rational to effectively zero, regardless of what else they had said up to that point; in my experience, it is not possible for a person to know enough about rationality to practice it, and simultaneously be ignorant enough of the natural sciences to believe that the Earth was created in essentially its present form with its present biota less than 10,000 years ago.

Being a conservative Catholic, as I understand that phrase, necessarily entails believing that homosexuality and contraception are morally wrong according to "natural law" which can supposedly be derived without recourse to divine revelation, and also believing that the College of Cardinals, a group of men who conspired to conceal the sexual abuse of children on a massive scale and thus enable it to continue for decades, are the best possible arbiters of morality for the rest of us. (If you don't believe those two things, you may still be a liberal Catholic, but you are not a conservative one.) Those beliefs are likewise not ones that someone can both hold and be a rational person. They do not, however, preclude intelligence; I would note Justice Antonin Scalia as an excellent example of a highly intelligent, deeply irrational conservative Catholic who uses his intelligence in the service of his irrational beliefs and goals.

Probably not, no. In general, it's just not that weak evidence.