If the South had won the Civil War, would we still have slavery in America today? Why, or why not?
Assuming that you are talking about racially-based, large scale slavery (as opposed to human trafficking by smugglers of illegal immigrants, which we do still have), we would not still have slavery in the US today, regardless of the outcome of the Civil War.
If the South had won, slavery would have died out by now for two reasons. First, it would have been condemned by public opinion around the world. It was such condemnation that drove Brazil to abolish slavery in the 1880s without a civil war. Countries around the world would have put pressure on the South to end the practice, just as countries put pressure on South Africa to end apartheid.
Second, slavery would have been destroyed by economic means. Slave plantations would never have been able to compete with large farms growing cotton using machinery and chemicals and such as is now done. The old slave system would have died out as agribusinesses around the world grew cotton that could outcompete the South's slave-grown cotton.
One way or the other, slavery was on the way out. It would have taken a lot longer, but it would have died eventually.
Absolutely not. Slavery was already on its way out around the world, and it is one of the prime reasons that England and other countries did not rush to support the Confederate States with troops and financial aid. The importation of slaves had already been made illegal in 1808, and the Southern states abided by this law. The British had completely outlawed slavery in 1833, and virtually every modern nation had abolished slavery by the turn of the 20th century. Brazil became the last American nation to ban slavery in 1888. In short, worldwide public opinion would have eventually forced the Confederate States to end their practice of slavery, even if the South had won the war. For an interesting look at this fascinating possibility, you may want to read the historical fiction novel, Guns of the South, by science fiction writer Harry Turtledove. In his novel, the South does win the war (thanks to a time machine that transports automatic weapons to the Confederate armies), Robert E. Lee becomes President, and Nathan Bedford Forrest takes command of the Confederate armies (instead of the Ku Klux Klan).