Before I start to answer this, some information you may find interesting:

The UN Partition Plan of 1947 gave an estimate of 145,000 Christians among the total population of ~1,845,600 in 1946. This placed them at less than 1/4 the Jewish number (608,000) and also meant that they constituted less than 10% of the population.

That's not to say that the Christian and Muslim Palestinians didn't get considered differently, however. Considerations differed in how each group was approached during the war, without a doubt, but not so much after the war. For example, Israel hesitated to attack Jordan in 1948, choosing to focus on Egypt after one of the truces had ended, because it didn't want to raise the ire of the Christians in the conflict by attacking holy sites in Jerusalem and Bethlehem. This was one of many reasons, but the fact that it was considered does explain that Israel was attempting to walk a very fine line.

Christians indeed acted as separate groups during the war in some capacity, as did Druze. The Druze actually ended up defecting from the Arab forces to the Israeli ones, allowing Israel to capture many Druze villages without opposition, and the Christians also contributed in their own ways to the fighting. Truces were organized in areas like the eastern shore of the Sea of Galilee, for example, between the Haganah (Israeli main army pre-IDF) and Arabs in the area. An example:

The harvest was cited by Druse and Christians in the Shafa ‘Amr-Ramat Yohanan area as a reason for not allowing or initiating hostilities.

Now, to get to your specific questions.

Was either group more permitted to stay in what became Israeli territory?

The question of "permitted" to stay is a contentious one in and of itself. Many argue still today that anyone was permitted to stay in Israel, but that the Arabs fled war, and only a very small fraction were expelled. Others argue that there was a centrally organized attempt to destroy villages to ensure Arabs fled. To what degree both are true is still contested. Morris argues that many Arab villages were destroyed by rather independently-acting field commanders but that many others fled because of fear, psychological warfare, and some small fraction were expelled directly. Karsh argues that more left due to fear, and being instructed to. Further, there's the question of cities like Haifa, Tiberias, and Jaffa. Morris paints the picture of tens of thousands of Arabs leaving because they couldn't accept an unconditional surrender, Karsh points to the Arab leadership telling the Arabs that they had to leave, even if they wanted to stay.

In that regard, being "permitted" to stay is still not really a question. Christian Palestinians who organized truces didn't have to fear war, but I don't think there's been a very straightforward study of whether or not they were more likely to sign non-aggression agreements with IDF forces. In terms of being permitted to return, I haven't seen any difference either. Many Christians (more than came, for sure) left Israel after the 1948 war regardless, it doesn't appear that there was a difference in terms of returning to Israel. In Israeli law, Palestinians were Palestinians, and were only defined separately very recently to my knowledge based on Christian or Muslim, though even that's contentious I believe (and recent, as I said). In Haifa, when negotiating in attempts for a truce, the National Committee in the area and the Arab Higher Committee couldn't come to an agreement on what to do. The AHC pushed for further fighting, while the NC was attempting to pursue a truce, though the AHC was supposed to outrank the NC since the NC was a branch of the AHC. This, happening in late January, was when most of Haifa's Arabs evacuated. Haifa Christians began to fear the worsening economic conditions and the continued skirmishes during negotiations, and many left, disheartened and demoralized. This seems to have been the poster child example of what happened throughout Israel. In Jordan, Christians felt (during the war) they were being overtaxed (particularly in the Old City) since the economic conditions were being worsened, while Muslims were paying bribes to get away with avoiding guard service and the like. Things like that, and Jewish intimidation, demoralized Christians and encouraged them to leave Israel/Palestine just as much as Muslims, though again I'm not aware of any firm count on that. However, Morris does say that in fighting, Christian villages were usually left in peace, because they were traditionally friendly, or not unfriendly anyways, to the IDF. Those villages that were taken were not usually subject to expulsions, and Muslim inhabitants fleeing didn't always cause Christians to do the same. It really seems like a case-by-case basis. After the war, 150,000 Muslim and Christian Arabs remained in Israel (the number of Christians alone that had been there before it began). Gandolfo claims 60,000 Christian Palestinians had left by the end of the 1948 war for Arab countries, alongside roughly 500,000 Muslim Palestinians. That would leave roughly 90,000 Christian Palestinians. He also estimates half resettled in Jordan (or Jordan-occupied territory), and Christians in Jordan did quite well, though less so in other Arab countries. As Prince El-Hassan bin Talal noted in 2001:

Christians across the region have done well, and any minority that has a successful record is looked at with a certain degree of envy.

In any case, it's less likely that more were permitted to stay, but rather that more were willing to and were willing to avoid conflict with the Israelis (with good reason, in some cases). Many still fled, however, regardless of the above, either because their neighbors did, fear, search to avoid a war, or because of Israeli/Arab actions.

Pappé does point out that both Christian and Muslim endowments were seized:

Neither were the Christian churches immune from this land grab. Much of the land that churches owned within destroyed villages was confiscated like the Waqf endowments, although unlike the vast majority of mosques, quite a few of the churches remained intact.

He also does point to the fact that in a few villages, Christians were given preferential treatment in staying if they were the ones under attack, unlike Muslims, as I said above. An example:

The villagers [of Iqrit], being Christians, were allowed to stay in the country, but not in their village. They did not capitulate, however, and began a protracted legal struggle for their right to return home, demanding that the army keep its promise.

Pappé does argue, however, that Christian villages were less cooperative than Morris explains. He says that while the Druze were cooperative with the Israelis when they realized which way the tide was going, the Christians refused any agreement by which they could stay in their villages for a short time if they pledged allegiance to Israel, and were thus treated just like Muslim villagers who gave that answer were. This is probably a case-by-case basis thing, as I keep saying, but I'm more inclined to believe Christians were given preferential treatment in more cases than not, especially later in the war (as Pappé himself admits) if only because Pappé tends to be more radical in interpretation and fails to contextualize/avoid assuming regarding his information more often than Morris (by most historian accounts).

So that should answer the first question!

Were the Arab states more likely to accept Christian or Muslim refugees?

To my knowledge, no. From the start of the conflict, the Arab League asked all nations to open their arms to all refugees. For example:

"The Palestinians had neighbouring Arab states which opened their borders and doors to the refugees, while the Jews had no alternative but to triumph or to die," Muhammad Nimr al-Khatib, a prominent Palestinian leader during the 1948 War, summed up his nation's defeat and dispersion.

Another example:

For his part King Abdallah reportedly promised that 'if any Palestine Arabs should become refugees as a result of the Husseini faction's activities, the gates of Transjordan would always be open to them'.

The distinction was not likely made, nor can I find any source that says it was. It's hard to prove that there was no distinction, because proving that something didn't exist is usually harder in my experience with history than proving something did, but I'm sure there were small incidents of discrimination. However, it also seems more likely that Arab nations accepted both in full where possible, and UNRWA worked to accommodate them equally in the camps (your third question). However, Palestinians were still marginalized in many societies, though Israel and Jordan both tried to "re-educate" them to tell them that they were Arab-Israelis or Jordanians, not Palestinians (both sides had their reasons for this). But again, I've seen no evidence of Christian-Muslim disparity in treatment. I'm sure some existed, as there tends to be tension between majorities and minorities in situations of scarcity like the refugee camps and their squalid conditions, but I can't seem to find any evidence that there was a systematic discrimination. Perhaps someone with more knowledge can come along and contribute more.

Sources:

K. Luisa Gandolfo. The Political and Social Identities of the Palestinian Christian Community in Jordan Middle East Journal, Vol. 62, No. 3 (Summer, 2008), pp. 437-455

Karsh, Efraim. The Arab-Israeli Conflict: The Palestine War 1948. Oxford: Osprey, 2002. Print.

Morris, Benny. Righteous Victims: A History of the Zionist-Arab Conflict, 1881-2001. New York: Knopf, 2001. Print.

Morris, Benny. The Birth of the Palestinian Refugee Problem Revisited. Cambridge: Cambridge UP, 2004. Print.

Pappé, Ilan. The Ethnic Cleansing of Palestine. Oxford: Oneworld, 2006. Print.