There never can be a completely definitive answer to the first question. Even a written confession by the hunter who shoved a spear into the last mammoth leaves room for us to question whether that really was the last mammoth or if he just thought it was. We can never completely rule out the possibility that aliens vacuumed up most of the mammoths and left that poor hunter to take the blame. Uncertainty is is an inescapable part of all historical sciences. We've learned to live with it and so should you.
The second question has some interesting possibilities. Even if we never manage to do it, the "not yet" answer can stand forever. Just as we must allow that there is a vanishingly small, but never quite vanished, chance that even the most ridiculous explanation is possible in history, we must also allow that, in technology, the impossible is only impossible because we're not doing it right. There is always a remote chance that new information or a new philosophical approach will show that everything we believe is wrong and that yesterday's "obviously, it's impossible" is tomorrow's "I can't believe they were so blind." We have already cloned living animals and living animals from frozen cells. In theory, that means the biggest problem in cloning a mammoth should be finding some intact cells to clone.
Mammoths could be revived in two ways. The first is simple in vitro fertilization. If we find a mammoth testicle (presumably attached to a mammoth, but you never know) that has a few sperm cells in it that were undamaged by freezing, we can use those sperms to fertilize an egg from an Indian elephant (genetically, the closest relative to a mammoth). We then impregnate the elephant and she carries the mammoth to term. The resulting baby is not really a mammoth, but rather a mammoth-elephant hybrid. With more mammoth sperm, we can impregnate the hybrid and produce ever more mammothy hybrids. Of course, if we are still working from our original supply of mammoth sperm, the hybrids will become increasingly inbred. In time we will have almost pure mammoth, but they will be really stupid.
The other plan is true cloning. For this we can use any cell from a mammoth. We remove the nucleus from the cell and use it replace the nucleus of an egg cell from an Indian elephant. We start the cell dividing by electrically or chemically shocking it. We then place the mammoth blastocyst in an elephant surrogate mother who gives birth to a genetically pure mammoth eighteen months later. Two variations on this method involve building our own mammoth DNA to place in the nucleus of an elephant egg. This is done by reassembling the fragments of DNA that we normally find in frozen mammoth tissue or by going over an elephant genome and modifying all 400,000 places where it differs from a mammoth genome.
As much as I would love to have my own pet mammoth, I know I will never see one. The big flaw in both plans is that we have never found any type of mammoth cells that weren't damaged by freezing, let alone fertile mammoth sperm. The two variants on plan B involve a level of genetic manipulation that is beyond our ability at the moment. Technologies of that sort always turn out to be harder to achieve than the sounded in theory. We probably will be able to assemble DNA bit by bit someday, but I strongly doubt I will live to see them assembling extinct animals using that method.
Some people worry about the ethics of recreating a mammoth. The mammoth's world is gone. The environment in which it flourished has vanished. I don't mean the ice age. Woolly mammoths evolved several ice ages ago and survived through several interglacial periods similar to the one we are living through today. What's missing is the mammoth steppe, the Arctic grassland that sustained mammoths. I'm less worried about that. Colombian mammoths lived in environments similar to those still found in parts of North America. Woolly mammoths should be able to adopt to one of those. If they can't, it is possible to recreate the mammoth steppe. For over twenty years, a Russian wildlife biologist, Sergey Zimov has been doing just that at a reserve in the northeastern corner of the Sakha Republic (that's Yakutia to you Risk players). The plan is, that by recreating an animal assemblage made up of analogs of the animals that lived there during the Pleistocene, the animals will do the work of selecting plants until the mammoth steppe has been recreated. Beyond simply being an interesting experiment, Zimov's Pleistocene park has the very real application of serving as a refuge for Asian animals that are endangered in their current range. Candidates include saiga antelope, Tibetan antelope, Amur leopard, and the Siberian tiger.
There is final reason why I'm quite sure I will never see a mammoth. Even if we manage to produce a genetically correct, living mammoth using one of these methods, it still will not be a real mammoth. Animals are a product of their environment. Genetics is only one factor in making an animal. Zimov's Pleistocene park ecology will only approximate the real thing. However, I won't make too much of a fuss over that. The mammoth steppe wasn't unchanging. Pleistocene park is close enough. My objection is that, however genetically correct a reconstructed mammoth is, and however close its environment is to the original, the nurture part of the nurture vs. nature equation can never be replaced. A reconstructed mammoth will have to be raised by an elephant. However well intentioned the elephant is, she can only teach the mammoth how to be a mammoth. Without time travel, we can never have a real mammoth; we can only have an almost-mammoth.
Not that I would turn one down.
Note: I was inspired to write this after reading a nice post that Brian Switek wrote about the first great mammoth cloning controversy. Go read it.