I'm not from US, so I don't have a "deep" knowledge of US history. Nevertheless- recently I've developed some degree of interest in how US developed. Thus, knowing that Dems/Reps are political parties which exist almost from the beginning of US, I was really interested in what was their stance on slavery, egality and so on. I was rather shocked when it occurred to me, that it were Democrats that supported slavery while Republicans had exactly opposite goals. When did it actually start then?
As an extra question, being only partially historical- what's the reason for modern branding of them? I can see overwhelming number of voices calling Republicans outright "evil" and Democrats as "good"- was it like that since these parties were founded?
It's a complex and long history, but the short version is that for various reasons Northern Democrats began supporting Civil Rights causes to various degrees in the 1940s and 1950s, and fully embraced them in the 1960s. This alienated Southern Democrats in particular. The GOP sought to take advantage of this by aligning themselves with southern Whites, "states rights" (to be racist), and being crypto-racist. This was known as the "Southern strategy" and it drove African-Americans to the Democratic Party in droves.
Obligatory Lee Atwater (GOP strategist) quote from 1981:
Atwater: You start out in 1954 by saying, "Nigger, nigger, nigger." By 1968 you can't say "nigger" — that hurts you. Backfires. So you say stuff like forced busing, states' rights and all that stuff. You're getting so abstract now [that] you're talking about cutting taxes, and all these things you're talking about are totally economic things and a byproduct of them is [that] blacks get hurt worse than whites. And subconsciously maybe that is part of it. I'm not saying that. But I'm saying that if it is getting that abstract, and that coded, that we are doing away with the racial problem one way or the other. You follow me — because obviously sitting around saying, "We want to cut this," is much more abstract than even the busing thing, and a hell of a lot more abstract than "Nigger, nigger."
If you want more details, Rick Perlstein's Nixonland: The Rise of a President and the Fracturing of America (2008) is a great read on how the GOP reoriented itself in the 1960s and 1970s, both on this issue, and its courting of the white Evangelical vote.
On your second question, obviously it varies by who is making the labeling, but if you believe that racism, sexism, and prejudice are absolute evils, and if you believe that the GOP has aligned itself with these positions, then it make it hard to see them in any other terms than this. Not everyone, obviously, sees them or these issues in that way. At various times, the parties have had strong moral sentiments attached to them that can elicit emotional responses. If you ask Evangelicals about the morality of the parties they tend to think the Democrats are "evil," for example, because of their support for abortions (which they view as an absolute evil, and Democrats generally do not). There is a large literature on polarization in recent US politics, but it is outside the scope of this subreddit.