These questions obviously could be answered at great length, but here I have to stay brief.
First, the easy one on finding ages of things: Of course 14
C is only useful for some 50,000 years, good for human history and some prehistory but not for most of the earth's history. To cover longer time periods the radiometric methods used include potassium-argon, uranium-thorium, uranium-lead, rubidium-strontium, and samarium-neodymium. (see http://en.wikipedia.org/wiki/Radiometric_dating
) All of these rely on the steady operation of the basic physical process of radioactive decay. They converge on an age for the earth of some 4.54 billion years. (see http://en.wikipedia.org/wiki/Age_of_the_Earth
Second, on evolution. I recommend the excellent book "Why Evolution is True" by Jerry Coyne.(http://www.amazon.com/Why-Evolution-True-Jerry-Coyne/dp/0670020532
) Here is an extremely brief summary of the principle lines of evidence.
1. Nucleic acid (and protein and anatomical) family trees. The relations between different organisms are evident from the similarities and differences of their DNA, particularly the parts of the DNA which change randomly over time and serve as a sort of molecular clock. The family tree of our relationships can be reconstructed in multiple ways from different parts of the DNA, RNA, and proteins. These ways show very good agreement with each other. There has been no explanation of this tree by any sort of creationist theory.
2. The fossil record. Dated fossils have been found for many of the most interesting transitions expected from the DNA trees. Among the most important transitions that are well-represented are reptile-mammal, reptile-bird, fish-amphibian.The dates agree nicely with those expected from the DNA. Even the locations of these fossils (e.g. for the fish-amphibian transition) were often predicted based on our understanding of evolution.
3. Biogeography. The locations of different species around the world are just what would be expected for life spreading by normal physical modes, then differentiating into species . Darwin's finches are one famous example.
4. Atavisms. Many organisms, including us, are full of internal remnants of no-longer functioning ancestral features. There are obvious things, like whales' tiny leg bones that occasionally develop into easily visible external legs. More numerous are the left-over genes, for example hundreds of non-functional slightly mutated genes in us for smell-receptors that were useful for ancestors.
5. Maladaptive features. We have many awkward traits which are obvious leftovers from evolutionary history and would make no sense for any sort of design. A painful example is the tendency toward hernias, due to the weird indirect path followed by the tubes carrying semen, which hook around arteries far out of the direct path.(http://www.scientificamerican.com/article.cfm?id=this-old-body
) The reason is that evolution does a very poor job of making discontinuous changes, e.g. topological changes, even when a child with a crayon could draw an improved design.
6. Observed evolution. Bacteria, viruses, and even insects evolve very rapidly. That's a huge problem, because it means that antibiotics etc. don't work well for long. Several new species of plants have been observed to start up within recent human history.
That should get you started, even though it's a tiny fraction of the total picture.
As for the part about some new set of rock layers just like the ones in the Grand Canyon but laid down quickly near Mount McKinley, so far as I can tell that's pure fiction.
(published on 11/20/10)