How did Christian Churches deal with segregation laws in the United States and are there any Bible passages that deals with discrimination and segregation? What were typical responses from Churches that were within that sub-culture? Were there Churches on both sides of the issue?
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:
The answer to your questions are thus:
"... are there any Bible passages that deals with discrimination and segregation?" Yes, James 2, https://www.biblegateway.com/passage/?search=James%202&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.