There were definitely many slaves who had sexual relationships with their masters. To pick one testimonial of very many, Mary Chesnut (wife of Senator James Chesnut of South Carolina) wrote in her diary about how the master of virtually every plantation slept with his slaves, but everyone pretended it didn't happen:

I wonder if it be a sin to think slavery a curse to any land. Sumner said not one word of this hated institution which is not true. Men & women are punished when their masters & mistresses are brutes & not when they do wrong -- & then we live surrounded by prostitutes. An abandoned woman is sent out of any decent house elsewhere. Who thinks any worse of a Negro or Mulatto woman for being a thing we can't name. God forgive us, but ours is a monstrous system & wrong & iniquity. Perhaps the rest of the world is as bad.

This is only what I see: like the patriarchs of old, our men live all in one house with their wives & their concubines, & the Mulattos one sees in every family exactly resemble the white children -- & every lady tells you who is the father of all the Mulatto children in everybody's household, but those in her own, she seems to think drop from the clouds or pretends so to think -- Good women we have, but they talk of nastiness tho they never do wrong; they talk day & night of.

Chesnut wrote this in the late 1850's, but this wasn't a new practice. As early as 1811, Elijah Fletcher wrote of Thomas Jefferson's relationship with his slave Sally Hemings:

That he cohabits with her and has a number of children by her is a sacred truth — and the worst of it is he keeps the same children slaves — an unnatural crime which is very common in these parts.

It's possible that some of these relationships were consensual to some extent. Sally Heming's son Madison later wrote that his mother voluntarily agreed to return with Jefferson from France. What we know, however, is that the master had almost every possible power of coercion over his slaves, and at least some - if not many - consciously took advantage of it. Harriet Jacobs of North Carolina, for instance, was sexually harassed by her master for years until she finally escaped and later wrote her story. Undoubtedly, she was merely one of many.

Absolutely. One of the famous examples that I can think of is Irish immigrant Michael Morris Healy, who settled in Georgia in 1819 and amassed a fortune. In 1829, Healy married one of his African-American slaves, Eliza Clark, and they had 10 children.

Eliza was a mulatto, and her children were so fair-skinned that they could pass for white without too much trouble. Healy knew this, and one by one, he brought his children north to New York, where they were cared for by a business friend. The girls became nuns (one later left her order to marry), and the boys attended school and went on to great success.

One was valedictorian in the first class to graduate from Holy Cross College in Worcester, Massachusetts. Another, named Patrick, went on to become president of Georgetown University.

The family's remarkable history is chronicled in Passing for White: Race, Religion and the Healy Family, 1820-1920 by James O'Toole.