Southern Democrats were more pro civil rights than Southern Republicans, and Northern Democrats were more pro civil rights than Northern Republicans. But Southern Democrats were opposed to civil rights as compared to Northern Republicans, and since there were a lot of Southern Democrats there were a lot of anti-civil rights Democrats.
No, this is false. In fact, during the Civil Rights movement, the majority of those in congress who voted in favor of reforms were Republicans. There was just one major exception, which was that Barry Goldwater voted against one of the Civil Rights bills, and he voted against it because he was libertarian and he didn't want to force it upon private businesses, e.g. "We reserve the right to deny service to anybody for any reason."
Being a libertarian myself, I agree with that viewpoint, but it has nothing to do with racism. If I owned a business, I wouldn't deny service over race. But I would deny it to a gangbanger who comes in with baggy clothes hanging so low that you can see the brown stains on his whitey tighties. I've actually that walk into a used car dealership I used to work for, and this person was asking to be financed for a used car, with the immediate answer being no. Yes he was black, but race wasn't the reason for it. Still, I'd be pretty pissed if some law forced us to offer a loan to somebody who comes in looking like he has probably never made a timely payment in his life.
At any rate, this particular opposition won Barry Goldwater the electoral votes in his home state as well as a number of southern states. However, the three presidential elections afterwards, none of the southern electorates went to Republicans. The first for that to happen (other than goldwater) was Richard Nixon, who took basically the entire nation (including left wing havens New York and California.)
You might want to read this, which consults several historians and has sources: