How have US Christian churches handled discrimination and segregation in the past?

"... are there any Bible passages that deals with discrimination and segregation?" Yes, James 2, https://www.biblegateway.com/passage/?search=James%202&version=NASB

Galations 3:28 https://www.biblegateway.com/passage/?search=Galatians%203&version=NASB

There is neither Jew nor Greek, there is neither slave nor free man, there is [aj]neither male nor female; for you are all one in Christ Jesus. 29 And if you [ak]belong to Christ, then you are Abraham’s [al]descendants, heirs according to promise.

If we replace Jew & Greek with black, white, brown, etc. the passage declares men & women of different races equal in God's sight.

I've been taught, the Hebrew culture of the time frowned upon men speaking to women in public. But we have several examples in the Gospels where Christ spoke to women. Many Jews In John 4 (see the same website), the woman at the well v. 8, 20 & 27 V. 8 sounds like a polite way of saying, many Jews shunned the Samaritans. So in that culture, she had too strikes against her, but Christ spoke to her & invited her to receive "living waters" irregardless of her race or gender.

Presbyterian.

Robert Bruce Mullin says in 'North America', published in A World History of Christianity (edited by Adrian Hastings), pages 437-8, that Southern religious figures claimed not only that was slavery a positive good, but that it was a Christian institution. He cites the Presbyterian theologian Charles Hodge (1797-1878), who insisted, “nothing is obligatory upon the conscience but what [the Bible] enjoins; nothing can be sin but what it condemns.”

The Southern Presbyterian Church finally began to address racism in earnest during the 1940s. In the late 1950s, a sufficient number of PCUS pastors supported the integration of Central High School, Little Rock, Arkansas, for Governor Orval Faubus to take note and label them communists. At the same time, the editors and many of the guest writers at The Presbyterian Journal supported segregation on theological and social grounds. The Journal's 1942-1966 theological case for segregation had four overlapping legs: the curse of Noah, divine approval of geographical segregation and disapproval of miscegenation, biblically-mandated cultural segregation, and Jesus's implicit support for segregation. The Presbyterian Journal changed its editorial policy regarding racism on 23 November 1966, when it published the complete text of "One Race, One Gospel, One Task," the statement of the World Congress on Evangelism. In 2002, the Nashville Presbytery proposed that the General Assembly adopt Overture 20, which recognised the sinful nature of racism, oppression, and exploitation then stated that the effects of these had divided and disadvantaged people. The overture also confessed collective sins: "As a people, both we and our fathers, have failed to keep the commandments, the statutes, and the laws God has commanded." It repented of "our pride, our complacency, and our complicity," and sought forgiveness from "our brothers and sisters." Finally, according to the Overture, the PCA should seek racial reconciliation and engage in interracial evangelisation.

Baptist.

Colonial era Baptists welcomed African Americans, both slave and free, allowing them to have more active roles in ministry than did other denominations by licensing them as preachers and, in some cases, allowing them to be treated as equals to white members. But by the Second Great Awakening, Baptist preachers had begun to interpret the Bible as supporting the practice of slavery and abandoned their pleas that slaves be manumitted. In 1995, the Southern Baptist Convention voted to adopt a resolution renouncing its racist roots and apologising for its past defence of slavery, segregation, and white supremacy.

Episcopal Church.

Clarence Bolden (Deputy, Diocese of Alaska) says the policies of the Episcopal Church of the United States of America (ECUSA) regarding slavery before, and segregation after, the Civil War years warrant an apology to blacks in America and in The Church. He says that even in the North, prominent members of the Church sought to justify slavery and racism. Dr. Samuel Seabury, a rector turned professor of the General Seminary of New York, wrote American Slavery Distinguished from the Slavery of English Theorists and Justified by the Law of Nature. In spite of amendments to the Constitution of the United States, racism after the Civil War persisted in the form of segregation. The Church tacitly approved segregation by neither condoning it, nor taking a position against it. Even the most liberal northern dioceses were unwilling to elect a black bishop if it meant elected black bishops would rightfully, under the Church law, have jurisdiction over both black and white churches. Under these conditions, black bishops could not even supervise black work. However, when members sought to propose special missionary districts and bishops for different races, the General Convention’s committee in 1877 “reported that it was ‘inexpedient to take any action in regard to providing Bishops exclusively for persons of different races and tongues.’” This advice was accepted by the General Convention. Bolden says that it was only around the 1940s with the annulment of the segregation of armed forces and civil services in 1948, and the Supreme Court ruling in Brown v. the Board of Education in 1954 that the Episcopal Church, although never having had a national segregation policy, “began to dismantle its institutional segregation policies.”

Catholicism.

Andrew S. Moore, from Saint Anselm College, says that most of Alabama's white Catholics shared white southerners' racism and initially opposed the goals of the Civil Rights Movement. For most of the civil rights movement, the Catholic Church in Alabama remained on the margins of the debates over integration and focused on internal Church affairs. It took the 1965 Selma to Montgomery march for voting rights, to draw the Church from the margins into the mainstream of the movement. After the mid-1960s there was little official sympathy for segregation.

The doctrinal controversies inherent have been discussed in Does the Bible support slavery and (white) superiority in the context of American slavery?, so I will limit myself to actual movements. This question: When did the African-American population become Christian? also talks about the rise of the black church.

Perhaps the most famous examples of how the church both engaged in and overcame racism are:

1. The emergence of the AME Church

   The AME Church was founded by Richard Allen, a black minister in Philadelphia in the late 1700s. Although licensed, his ministry was restricted to blacks, and was relegated to early morning services. When one prayer meeting went long, the story goes, the whites came in and told him he had to move, even while in the midst of prayer. When he demanded that the prayer should continue, the white men beat him savagely. This was the genesis for the denomination to begin.

2. The abolitionist movement, and William Wilberforce

   Slavery was abolished in the United Kingdom in 1807, and William Wilberforce is most commonly credited with the legislative victory. A committed Christian, he understood that God was no respecter of persons, and that in Christ, there is neither Jew nor Greek, Slave nor Free. The Abolitionist Movement in the United States took largely the same queue.

3. The emergence of the Civil Rights Movement amongst churches in Alabama

   That the Southern Christian Leadership Conference was started by the Reverend Dr. Martin Luther King, Jr. is not a coincidence. Much of the early organizing against segregation was coordinated in churches in Montgomery, Birmingham, Atlanta, and later across the south.

4. The Apology By The Southern Baptist Convention for Slavery in 1995.

   The largest denomination in the United States outside of the Roman Catholic Church is the Southern Baptist Convention. It was formed in 1845, as the result of a split between the Northern Baptists and Southern Baptists, over the question of whether or not missionaries should be allowed to keep slaves. In 1995, the SBC acknowledged this history, and asked forgiveness for it.

The answer to your questions are thus:

• The church was on both sides of the issue, perpetuating evil and good in this regard. (I think I can get away with calling discrimination and slavery as evil).
• In general, the White church tended to justify itself, but later came around on the issue
• The Black church in the US is neglected way too much.