Can Christians who share some ethical and political values with Rand be Objectivists?

No. The essence of being Christian is that you believe that Jesus was the son of God and that he was sent to Earth for the salvation of mankind; at least, I'm pretty sure every sect of Christianity would agree with that much. Objectivism is a strictly atheist philosophy which holds that there is no God. These two views are mutually exclusive.

No.

The philosophy of Objectivism does not hold a monopoly on people of virtue.

Being an Objectivist isn't synonymous with being good.

While it is true that, to truly be an Objectivist is to be a good person, it doesn't follow that all good people are Objectivists.

The evidence for this is plain, and all around you. In a world where most people have never heard of Ayn Rand, and many, even most, people are essentially good, the answer's clear.

To be an Objectivist is to be in fundamental agreement with the philosophy of Ayn Rand, to agree with the essentials of the philosophy.

A few examples, of which there are countless more:

Metaphysically and epistemologically, accepting something as basic as the Law of Identity is a tall order if you also believe in any one (much less all three) of the impossible concepts of omnipotence, omnibenelovence and omnipresence.

Ethically, to believe in Original Sin and vicarious redemption through the sacrifice of another is a denial of any rational view of morality.

Politically, to render unto Caeser that which is Caeser's and unto God that which is God's, is a denial of Individual Rights, Capitalism, and the United States of America.

The most I could allow is, regarding a Christian Objectivist, I'd say they could be an Objectivist to the extent that they're not an Christian, and vice versa.

Ayn Rand's correspondance indicated that she didn't place much emphasis on judging a person based on whether they were religious or not--her primary concern was whether they approached the issues of their practical daily life in a rational, egoistic manner. For example, I remember one woman was cited as an exemplary "intellectual activist" in her magazine, for speaking about the benefits of Rand's ideas to her church group. If Ayn Rand had contempt for Christians, or considered it as "Objectivism or Christianity but not both!" then she never would have cited that woman.

I think the most important thing about a person is not necessarily what they explicitly say is their philosophy, but rather, what is their actual philosophy based on their actions. I will take an intellectually honest, rational, rights-respecting, civilized person who may not be "an Objectivist", over some of the shrill "doctrinaire Objectivists" I have met over the years.

Objectivism is more than a set of principles; it is the integration of those principles into a complete and consistent system. If any of the principles, particularly the ones at or near the top of the hierarchy, is rejected the entire system fails. The loss of reasoning power that the principle supports, as well as the errors that gave rise to its rejection, will leave another of the principles without a valid defense, which will in turn undercut another, and so on.

Anyone who accepts the existence of a God, or any other mystical idea, has accepted an arbitrary assertion and has therefore rejected the need to ground ideas in the observable facts of reality. Once the relationship to reality has been severed, any conclusion can be drawn, including the one that the rest of the principles of Objectivism are invalid. This is why mysticism (of which Christianity is one form) and Objectivism are mutually exclusive.