Height isn't going to make any real difference for shortwave reception unless you're seriously high up and putting up a very efficient antenna like a dipole. At that point it's kind of moot, as you don't need an efficient antenna anyways to receive shortwave.
Correct about altitude... it is a common misconception that height always improves reception, because it doesn't take into account that radio waves propagate in vastly different ways according to their frequency and wavelength. Thankfully there are organizations like the IRAU and ARLL that have vast amounts of info on what does and doesn't work in a wide variety of locations, bands, and altitudes:
It's not a bad choice, but the tiny tecsun radio I have has a telescopic antenna that is about 3 ft long when extended. It receives shortwave very well.
If there is some distant broadcast (maybe overseas) that you want to receive you can definitely build a very efficient antenna system for that broadcaster's frequency. Be prepared to shell out well over $10,000 for this.
Regarding simply stringing long wire, if you have a huge, flat land area and don't mind overhead wires, erect a rhombic antenna on dedicated poles for extremely high gain but with a few downsides worth reading about, such as the sheer size needed for shortwave. Don't string wires between trees as they are guaranteed to snap in the wind.
as a ham, I highly recommend K9AY receiving loop antenna. you can find instructions online. it's a loop antenna design that's steerable via DPDT switch and has directional noise rejection. it also takes up less space horizontally than an efhw.
I tried that with my buddy's shortwave while on guard duty in Iraq and fried the thing due to atmospheric electrostatic energy discharge (or it's actual scientific name), needless to say he was less than happy with me. First clue probably should have been the visible sparks coming off the antenna wire. Second clue should have been I was sitting in a truck insulated from the ground. Yep, he definitely should have known better.
its more likely that the wire coupled in a very large amount of RF power from a transmitter that was at the base, damaging the unit. I obviously don't know your units deployment equipment, but the transmitter at the base is often greater than 100 watts of output
Could be, I always just assumed it was from the high prevalence of static electricity in the desert. When I lived in Phoenix I would always touch the back of my hand to anything metal (like doorknobs) before touching it with my fingers because I've been shocked way too many times.
An antenna extension (https://www.dxengineering.com/parts/sgn-ant-60) would help.
Even better would be an active antenna. I have only heard great things about the MLA30+ though I don't own one myself.
WWCR (4840) has always been the easiest broadcast for me to pick up in the US.