Yes, they had sex. Drouet was keeping Carrie as his mistress in an apartment. Probably everybody thought they were married. The plain fact was that Carrie decided it was easier to be a "kept woman" than to support herself by doing grueling work in a factory. The reason this may not be clear is that these things could not be spelled out explicitly in any stories or novels at the time Sister Carrie was published. I believe the publisher's wife read the novel in manuscript and was shocked by the story, as innocuous as it was. She wanted her husband to renege on his commitment to publish the book, but he was forced to bring out a small edition under his contract with Dreiser. The novel did not do well because it received no advertising or other promotion. This was in 1900. When Carrie ran away with Hurstwood there was no mention of sex either. In fact, if I remember correctly, there was no explicit mention of sexual activity even in Fitzgerald's The Great Gatsby, a novel about the Roaring Twenties when women were supposedly becoming liberated.
Yes, they did. The word Dreiser uses is "yield" which means that she submitted. What makes Carrie's sexual desire unique, as Sybil B. Weir argues in her article, "The Image of Women in Dreiser's Fiction," is that Dreiser "accept(s) the fact that women have erotic desires and to assert their sexual careers does not automatically invalidate their moral natures.