Short answers, pulled from the story.
Neanderthals went extinct roughly 40,000 years ago, coinciding with the immigration of modern humans into Europe. Populations in southern Spain and Gibraltar may have persisted until 32,000 to 35,000 years ago, and some Neanderthals in Gibraltar may have survived thousands of years longer than the broader extinction event.
The first recognised Neanderthal fossil, Neanderthal 1, was discovered in 1856 in the Neander Valley, Germany. Local schoolteacher Johann Carl Fuhlrott identified the bones as distinct from modern humans and gave them to anthropologist Hermann Schaaffhausen for study in 1857. Earlier finds, including Engis 2 recovered in Belgium in 1829, were not recognised as Neanderthal remains at the time of discovery.
Yes. The first Neanderthal genome sequence, published in 2010, confirmed interbreeding between Neanderthals and modern humans in at least two episodes outside of Africa: one roughly 250,000 years ago and another between 40,000 and 54,000 years ago. Any person with ancestry beyond sub-Saharan Africa may carry approximately 2% Neanderthal DNA, and roughly 20% of the total Neanderthal genome appears to have survived in the modern human gene pool.
Neanderthal brain volume averaged 1,640 cubic centimetres for males and 1,460 cubic centimetres for females, larger than the averages for all living human populations. The largest known Neanderthal brain, from the specimen Amud 1, measured 1,736 cubic centimetres, one of the largest ever recorded in any human. However, Neanderthal brain organisation differed from modern humans in areas related to cognition and language.
Neanderthals produced Mousterian stone tools using the Levallois technique and manufactured wood spears, including the 400,000-year-old Clacton Spear and the 300,000-year-old Schöningen spears from Germany. They also invented the earliest known dry distillation process to produce birch tar, a technique not documented in modern humans until 20,000 years ago, and may have used birch bark tar to haft spear tips.
Neanderthals did bury their dead, though the practice appears to have been uncommon. Burials were typically simple, shallow graves, but children's graves were often associated with grave goods including artefacts and bones. One grave in Shanidar Cave, Iraq, was found with pollen from multiple flowering plants, including yarrow, grape hyacinth, and hollyhock, though whether the flowers were deliberately placed remains debated.