So just try to do your best to enjoy these articles. Reading
the abstract, first paragraph, and closing paragraph is
usually a quick and practical way to get a sense of what are
the key ideas and implications.
The article ends with Feynman offering two personal rewards of $1,000 (a lot of money in 1959) to the first person to store a page of information on an area that was 1/25,000 the size of a book page, and to the first person to create an electric motor that can operate in a cube of size of 1/64 in. This latter challenge turned out not to be so hard with 1950's technology, and Feynman had to pay the motor prize off just a year later, in 1960. The first prize had to wait until 1985 and was paid to a Stanford graduate student who used electron beam lithography to reproduce the first page of Dicken's novel A Tale of Two Cities on a page measuring 1/160 mm on a side.
Roughly speaking, the speed of a pulse of light propagating through a dispersive material is described by the group velocity, which depends both on the index of refraction as well on as the derivative of the index of refraction with respect to frequency. When the refractive index of the material is a decreasing function of frequency, the group velocity can take on values that are greater than the speed of light in vacuum or even be negative. Gauthier's and Boyd's research groups have undertaken experiments that demonstrate this unusual behavior. Also, Gauthier's group has done experiments to show how such behavior is consistent with the special theory of relativity. The key is to understand how information is encoded on an optical waveform - the special theory places a speed limit on the speed at which information travels, not on the speed of a pulse. For additional resources, see this tutorial.
Briefly, it is possible to arrange two particles, say two photons arising from the annihilation of a positronium atom, to be entangled which means that these particles have behaviors that are correlated in a way that makes no sense from a classical (non-quantum) point of view: measurements on one particle of an entangled pair must produce results that are correlated with measurements carried out on the other particle of the entangled pair, even if the two particles are separated by a great distance (say a light year) and even if the measurements are simultaneously carried out on the two particles, more quickly than light can travel between the two experiments. Read the Mermin article to get a better understanding.