Did the Norse gods have children with mortals/humans as in Greek Mythology? Who were they? What did they do?
They mostly had children of their own.
The most famous such coupling is that of Rígsþula, which tells the story of how Heimdall, calling himself Rig, visits the homes of three different human couples, and sleeps in the same bed as them. After nine months, a boy is born, of which we are told a bit of his history, his marriage to a suitable girl, and the names of their children. We are told from these godchildren, the lines of thralls, of the freemen, and of the lords came.
Other sources, more historical or genealogical, trace the ancestry of specific lords back to the gods. Most known of these is perhaps that of the Ynglings, the kings of Sweden, with a cadet branch in Norway, who traced their line to Frey. We know this from Ynglingatal, and Snorri's retelling in Ynglinga saga in Heimskringla. Similarly, the jarls at Lade are in Háleygjatal said to be descended from Odin and the giantess Skadi. The Skyldings in Denmark also were said to be descended from Odin, as were the legendary Völsungs.
Of those born directly to the gods, I only know any stories about Fjölnir, first of the Ynglings, and Sigi, the ancestor of the Völsungs, and they are short: Sigi was banished for murder, and became king in the land of the Franks or possibly in that of the Huns. Fjölnir instead ruled his kingdom in peace, and on a visit to king Frode in Denmark, drowned in a vat of mead.