1) Not at all! There are many applications and examples not needing three PhDs to understand. Gosh, where to start...
2) Too many to list here, and I'm not expert, but one important general method is depositing layers onto a material, for example how transistors and other semiconductor devices are made. Vapor deposition, ion bombardment, and more. Application of voltages to the finished device alters the potentials in various ways, such as controlling the slope of a quantum well, which affects how electrons behave in there, which influences the device's interaction with light. Quantum wells are such a general idea, and certainly not confined to semiconductors - one could write a an impressively big book covering all the ways particular quantum potentials are made, and square wells are pretty common.
3) The steady progress of manufacturing technology reached the point where interesting applications of square well potentials were possible. Anytime something interesting becomes possible, it happens - there are thousands of EEs and physicists and creative inventors itching to make all kinds of ideas become real. Usually there's not any one thing in science that can be praised as the key to an invention; so much is manufacturing technique and improvements in lab equipment.
There are quantum dots, made by the same basic processes used for all semiconductors, etching and depositing layers, with whatever sophisticated refinements only the experts can explain, to make tiny cylinders or cubes standing on the substrate. This couldn't have been done decades ago because to have interesting quantum effects, these cylinders must be small, either comparable to optical wavelengths or smaller to affect electron waves.
There are also nano particles that are spherical, maybe coated with another material, with fun quantum well phenomena involving electrons, energy levels and optical properties.
Semiconductor manufacturing has been in a long uphill climb to make smaller and smaller structures so we can have faster CPUs, more memory, and all the rest. Every year or so, something becomes possible that had been theory. (I'm still waiting for Star Trek transporters.) If you really want to know much about the answers to your questions, you'll be reading history or chasing a moving target.
Definitely, read up on electronics. Here's an article that might be informative: http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_3.htm Google for "quantum well lasers" and for "two dimensional electron gas" which is a swarm of free charge carriers in a conductor confined to a thin layer due to confinement along the third dimension by a manufactured potential well. An important application of 2DEG is in the design of HEMT (High Electron Mobility Transistors - see http://en.wikipedia.org/wiki/High_electron_mobility_transistor) - essential in advanced radio astronomy and used at facilities such as ALMA. Other topics to pursue: metamaterials, blue LEDs (http://www.jpsalaser.com/apps_led.html), QWIPs (Quantum Well Infrared Photodetectors - try http://cqd.eecs.northwestern.edu/research/qwip.php or this PDF of research for the US Navy www.dtic.mil/dtic/tr/fulltext/u2/a275745.pdf has some juicy details, though that may be a matter of taste)
This article I think might be right-on in partially answering some of your curiosity: http://www.sandia.gov/media/quantran.htm (Quantum Transistors at Sandia Labs)
I don't know if this is of interest, but I feel an urge to mention: spectra and other interaction properties of organic molecules with double bonds, especially multi-ringed ones, "aromatic" molecules, can be understood to some extent by thinking of the valence electrons as stuck in quantum well boxes. Look up "Hückel method." By altering side chains, chemists can tweak energy levels to alter colors of dyes. I couldn't find a well-written article right now.
Sorry for this short uninformative answer - I'm lazy and not an expert! :P