I might not be able to get as many college-trained scientists to agree, but my answers are more accurate, despite that. On the PBS website they have a "FAQ about Evolution" that provides hyper-rationalist answers that really bug me, so I'm going to provide my own answers here.
1. Did we evolve from monkeys?
2. How did humans evolve?
Through natural and narcissistic selection, as did all other species of organism in biology. The apes who became human ancestors developed into a large number of species collectively referred to as hominids. A hominid is any human-like ape, but only one species of hominid was the direct ancestor of all modern humans, and only one species of hominid was the direct ancestor of that one ancestor hominid, and only one species of hominid-like ape was the direct ancestor of all species of hominids, etc. All of the other (non-direct ancestor) kinds of hominids were merely human-like apes, and the point at which we change our terminology from "ape" to "hominid" to "human" is one of philosophy and personal preference rather than science. Some people believe that most or all recent hominids are "human", some people believe that only our modern species (Homo sapien sapien) is human. Even what makes the important distinction either distinct or important is a matter of preference; some people believe that only neurological differences distinguish humans from other animals, while some people believe that other physiological indications are acceptable for distinguishing humans from other hominids or hominids from other apes.
3. Is culture the result of evolution?
The existence of culture is the result of biological evolution, specifically the neurological evolution which caused the development of language. Other than the fact that culture exists, however, the details of culture are not biologically determined, and so any particular aspect of culture is not the way it is because it "evolved" to be that way, but instead is the way it is because humans chose it to be that way, an intellectual selection whether conscious or not. The fundamental distinction is caused by the ability of humans, using the selection mechanism of reasoning, to consider alternative selections, requiring a capability of foresight and hindsight that is not available to the natural or narcissistic selection mechanisms of biological evolution.
4. How are modern humans and Neanderthals related?
The particular relationship, and even the identification of species, within the family tree of all hominids is very uncertain. Popular theories backed up by quantifiable data indicate that Neanderthals did not constitute a distinct species from other (more direct) human ancestors, as interbreeding between the two putative gene pools has left some genes which originally evolved in the Neanderthals in the current gene pool of modern humans. (In most cases, the ability to breed is considered definitive of a species.) Currently, the biological and environmental relationships between several kinds of "proto-human" populations and our own species is almost entirely uncertain. How the creatures we refer to as "Neanderthals", "Devonian" and Homo Erectus interbred cannot be easily analyzed, but the mere fact that they did interbreed, combined with the difficulty inherent in using both the words "species" and "human" with any deductive certainty, makes the reconstruction of human evolution at this time a matter of personal conviction rather than scientific knowledge.
5. What do humans have in common with single-celled organisms?
Like all other multi-celled creatures, humans are genetically related by ancestry to single-celled organisms. Prior to the evolution of species of colonial organisms (collections of essentially identical cells), which eventually evolved into species of somatic organisms (creatures with specialized cells), one individual single-celled organism (analogous to what we would today call a bacteria, known as a prokaryote) engulfed, without destroying, another single-celled organism. Both organisms continued to live, grow, and eventually reproduce, in tandem. Eventually evolution caused the ingested organism to specialize in being the 'genetic storehouse' of the combined 'creature', becoming a 'nucleus', while the engulfing organism lost all of the free-floating chromosomes that bacteria use for heredity, thus establishing a new kind of single celled creature known as a eukaryote; a cell with a nucleus. As biological evolution continued, one of the descendants of this first eukaryote in turn engulfed another prokaryote, and as they developed in tandem the population of descendants from the engulfed prokaryote evolved into mitochondria, an important (but separately reproducing) organelle within almost all eukaryotic cells. Some descendants of these cells in turn evolved into colonial organisms and eventually somatic organisms. (An individual eukaryote similarly engulfed a bacteria capable of photosynthesis, and this cell became the ancestor of modern plants, with the ingested-but-not-digested bacteria evolving into the organelle called a chloroplast.) Meanwhile, of course, all of the other prokaryotes that were never 'colonized' by these proto-organelles continued to replicate through the eons as the ancestors of modern bacteria.
6. What happened in the Cambrian explosion?
Based on the available fossil record, it appears that the Cambrian period (between 540 and 485 million years ago, a span of about 55 million years) was a time in which a huge diversity of different kinds and species of somatic creatures were evolving. Because their fossils do not preserve any genetic information, the hereditary relationships between the kinds, species, and individual organisms (phylogeny) cannot be known with any certainty, nor can the specific relationship between Cambrian forms and modern organisms. However, the development of the different types of animals we think of as 'distinct', birds, fish, mammals, etc, did not occur until long after the Cambrian 'explosion'. This means that they (and we) have all evolved from a single species of creature that survived the Cambrian, but we don't know precisely which kind of creature that was. Most of the examples from the Cambrian explosion can be excluded from consideration because their "body plan" is profoundly different from what we would expect such a common ancestor to be like.