In short: expect damage like that from surges or lightning, albeit somewhat more severe and much more widespread. I'll be answering your questions in turn, with the help of this resource that Nex Terren dug up and some EE knowledge of my own.
All surge damage, first off, is erratic in nature -- electrical breakdown is often a stochastic phenomenon promoted by impurities in materials, surface contaminants, or pre-existing subtle damage. I'd expect spot surge damage in a variety of places, some seemingly distant from others yet related electrically or mechanically (by arc blasts between mechanically nearby parts), and in components both mundane (discrete parts and PCB tracks) and critical and complex (such as ICs/chips). However, a total meltdown to the point of unrecognizability as electronics is unrealistic as the materials used as insulators in many parts are designed to withstand being baked in ovens, and just won't melt/burn up unless you put extreme heat into them -- even severe burn damage to circuit boards will not render them unrecognizable, and semiconductor mold compounds will split and crack under high energies far before they even smolder.
Damage from surges can take a variety of forms, and much of it depends on how breakdown-resistant and surge-current-resistant the various paths and loops in the system in question are as well as the size of the surge. A surge can damage the PCB and tracks on it as well as current-sensitive components such as resistors and inductors via excess I2t, overvoltage parts vulnerable to dielectric or avalanche breakdown such as capacitors and semiconductors, and burn and pit contacts through air arcing.
Smaller loops and shorter lines have an advantage in that they pick up less energy from the environment, but smaller feature geometries on ICs especially but also in discrete parts tend to fail at lower voltages and currents, counterbalancing this and rendering modern equipment more vulnerable than say something from the 70s built using discrete transistors. In addition, dedicated suppression components such as fuses and MOVs are likely to fail first, and in a way that stops further energy flow (fuses blow, MOVs fail short) -- these sacrificial failures may help by absorbing the brunt of the surge energy incoming to the device, especially if it's being conducted in by long lines.
Spares vary in vulnerability -- ESD-protection (especially conductive vs dissipative) bags or boxes may act as something of a shield, while ESD foam on the other hand can increase the vulnerability by closing loops that would otherwise not flow current. Boards are also more vulnerable than loose parts, especially for current-sensitive parts such as resistors, inductors, and of course fuses. However, spare boards may be less prone than live equipment to exhibiting certain destructive failure modes that require an existing energy source, such as SCR latchup in junction-isolated integrated circuits. Likewise, being unplugged has the advantage that you aren't vulnerable to surges coming in on long lines such as the AC mains or any sort of telecommunications wire.
Repairing EMP damage
All types of surge (ESD/static electricity, fast transient/inductive transient/small arc transient, lightning, geomagnetic surge, and EMP) damage follow a similar repair process: find all the bits that fried and replace them. However, it's the troubleshooting that's a pain in the arse due to the scattershot and often seemingly spooky nature of surge damage. A recent mikeselectricstuff video demonstrates this vividly as a microwave oven he received was rendered nigh-unrepairable due to a chain of events that started with an expiring 12V incandescent lightbulb and ended in a fried controller chip on the front panel.
Your intrepid technicians would be doubly challenged as they would have to test their spare parts to make sure they aren't putting dead or badly damaged parts in. For discrete components, this is mostly feasible; however, most ICs can't be comprehensively self-tested without custom test fixturing at a minimum, and sometimes require expensive test equipment to test at all.
The silver lining is that ICs are almost universally equipped with I/O protection structures on their pads, which will absorb the brunt of the surge energy and fail first, usually as opens or dead shorts. This makes it possible to "go/no-go" test the ICs for surge damage by testing all the I/O protection structures. If they all pass with flying colors, the chip stands a good chance of functioning at least mostly as designed, although analog parts are vulnerable to low-energy surges causing parametric shifts. A failed protection device, however, is an auto-toss, as that means the pad connected to that device is shorted out or no longer protected.
What survives, what doesn't
Most consumer electronics is liable to fail -- it's simply not designed to reject EMI well to begin with due to cost pressures, and an EMP is the ultimate expression of an EMC event. Well-designed quasi-mass-market gear (such as test equipment, two-way radios, or industrial controllers) is somewhat more likely to survive as it has better protection and shielding than bottom-dollar mass-market stuff; older equipment that relies less on IC technology is also more survivable than highly integrated stuff.
Appliances (white goods, HVAC, ...) are liable to have their fancy control functions and power controllers (inverters/...) fail, while the guts of the appliance stay mostly intact -- simple electromechanical control (relay, cam-timer) stuff has a good chance of surviving though, and is dead simple to fix, comparatively speaking, if it does break. Likewise, vehicles are likely to at least somewhat survive as they are heavily ruggedized electronically speaking and don't suffer from long-lines threats -- and even if the fancy computers fail, there are at least some fallbacks designed in to let you limp home or at least pull off to the side of the road. Aircraft will be even less plussed by this, especially if the weather's nice -- military/aerospace equipment is heavily ruggedized as well, and surprisingly few aircraft absolutely need electronics to fly in visual flight rules conditions.
As to components -- the major problems for component availability will be custom parts (normally ICs). This is a massive problem for mass-market gear, which relies heavily on custom ICs (whether it be a custom ROM masked into a microcontroller or a full custom ASIC). Specialized equipment is somewhat affected as well, especially if it relies on high-performance ASICs to achieve its level of functionality (vs. using more standard ICs along with processors or FPGAs for custom functions).
Jellybean parts (such as discrete parts, connectors, and multi-sourced standard catalog ICs), though, will still be likely to be available in part stocks and sometimes in equipment, even if they are not as available in junk as they were even 20 or 30 years ago. Hardened gear, of course, will also be likely to survive wholesale.
Chaos, but not catastrophe
Given all that -- I'd expect emergency communications to rebound quickly albeit in improvised form as radio amateurs can build working transceivers out of jellybean and hand-made parts relatively readily even to this day, and run them off of batteries. Vehicles would be in various "limp modes" depending on the type and age of vehicle -- the worst widespread case would be a car that won't run. Even many aircraft would be serviceable in visual flight rules conditions provided you had fuel. Of course, getting fuel would be a problem -- the pipeline/terminal infrastructure would be mostly out of commission due to dead pumping systems (you can throw people and improvised comms at the lack of a SCADA system, but the pumps need power to run), forcing fuel to be railed, barged (yes, barged), and trucked around. Under the circumstances, refineries would be dead, but there would be enough fuel in tanks that you could get it into railcars or trucks in a pinch by way of gravity.
Scientific and medical machinery would vary in damage depending on sophistication (the more sophisticated it is, the more damage it'll take), while the impacted parts of our computing infrastructure would take some time to rebuild, along with the damaged parts of the packet-switched terrestrial IP backbone. Satellite comms and navigation, though, would come back relatively quickly to at least some extent as geosynchronous and medium-orbit birds are hardened against ionizing radiation and EMI and satellites don't have long lines hooked up.
As to powering this all? Generators themselves are still likely to be somewhat functional (you'd need to bypass or kludge around electronic controls on modern generators, but old/simpler ones are likely to be OKish or at least readily fixable) even if the grid itself suffers geomagnetic-type damage, and most power operators are at least somewhat familiar with the type of threat a geomagnetic storm poses, which'd give them some basis for responding to a wide-area contingency like this. Furthermore, mobile generation capacity is available wherever the rail network on your continent goes -- older diesel locomotives are relatively well-positioned to shrug off EMPs (unlike the Soviet experience in 1962, there are no long wires connected to the electrical system on a locomotive to aggravate EMP problems), and can easily be turned into mobile emergency gensets that could be used to supplement existing hydroelectric plants for black starting grids. Another possibility to deal with some grid damage problems (large transformers taking geomagnetic damage, to be precise) would be to rewind the transformer onsite to restore a limited degree of functionality until spares could be made -- the likely geomagnetic damage modes are heating based due to the sluggishness of the pulse involved, and the insulation and oil in a transformer go kaput far before the copper itself is ruined beyond reuse.