Excellent questions; I'll take these out of order, forgive me here.
"The congregation of God" here simply means marrying into the Jewish mainstream. So a man who chose to be castrated is allowed to marry a convert, but not a born-Jew. "What is the spiritual significance?" Don't go there. It's a technical law on the books. For all other aspects, he's a Jew like anyone else. No one needs to know anything about this person's private parts, except for a woman he's considering marrying! The commentaries (e.g. Sefer HaChinuch) suggest that in Biblical times, a person could make good money and honor by choosing to be castrated (required for service in many royal palaces), so the Torah had to institute some sort of penalty to discourage it, effectively by making finding a partner significantly less convenient.
There is no analogous prohibition for non-Jews. (The Talmud debates whether non-Jews are allowed to get castrated, but there is certainly no penalty clause prohibiting categories of marriage if it was done.)
There's a contemporary text on Jewish law vis-a-vis medicine, Nishmat Avraham, that deals with your other questions exhaustively. Here's a link to an English version of the book. As to what counts as injured, this gets technical and graphic so I'll refer you to the above text. (But the Talmud says explicitly that the prohibition applies to someone who got their vas deferens damaged, too.)
In short, the prohibition is only on getting married, not staying married; hence if Joe Mainstream Jew marries Judy Mainstream Jew, and a few days later Joe is in some accident that damages his private parts, he's allowed to remain married to Judy.
And the Talmud says there is no penalty on someone whose anatomy was damaged by disease; I don't know the details but if I recall correctly this is applied as well to someone who had a testicle surgically removed because of cancer.