Factors can affect a magnet’s strength include heat, radiation, strong electrical currents in close proximity to the magnet and the presence of other magnets. Neodymium magnets will also corrode in humid environments without a protective coating.
No. Once a magnet has become magnetised it cannot become stronger. It's like filling a bucket of water, once the bucket is full it cannot be filled any more. To get a stronger magnetic attraction you will need to change the magnet to a larger one, or combine two or more magnets together.
Neodymium magnets are hard, brittle and easy to break or damage. They can also become demagnetised by the heat produced when machining. If you need to secure a magnet it's best to use adhesive or use magnets with holes for countersunk screws. If you want machinable magnets try AlNiCo magnets.
All neodymium magnets are graded with a letter: 'N', for 'Neodymium', followed by two numbers. These numbers represent the maximum energy product of the magnet, with the higher the grade, the stronger the pulling force of the magnet. Grades begin at N30 (lower grades are no longer produced) and end at N52. Strength is not the only important factor in choosing a magnet for your application, higher grade magnets are also more brittle and have lower maximum operating temperatures.
Not exactly, "Rare Earth" refers to a couple of types of permanent magnets made from special alloys: both Neodymium-Iron-Boron (known as Neodymium) and Samarium Cobalt (known as SamCo) are rare earth magnets. The elements that make up these allows were once thought to be rare, hence the name, but which we now know are found relatively abundantly.