Get a programmed-start ballast.
Cheap ballasts also misuse ground to manipulate capacitive effects between the fluorescent tube and the metal enclosure. Note huge assumption about fixture design.
Programmed-start ballasts preheat the fluorescent lamp filaments (in the ends of the tubes) until conditions are ideal to strike the arc in the manner Tesla intended. It results in a 0.5 to 3 second startup delay (depending on ambient temperature) but is very easy on the tube, and avoids the capacitive trick.
Heat will be minimal on any modern ballast. In a fluorescent, the heat comes from the tubes.
You may want to think about LED lighting and do an end-run around all these problems.
You can't use NM cable as cordage. If you want cordage, buy cordage. They often sell it with a socket premolded onto the end, marketed as an "extension cord". Make sure the writing on the side of the cord is legal for cordage, then SNIP!
This can't possibly be safe can it?
Most people outside the UK consider UK plugs and sockets to be, if anything, over-engineered for safety¹. The use of fuses, shutters, sleeved pins, polarised connectors, angled cable outlets, high insertion forces, finger protection and so on are not universal and ubiquitous outside the UK. Of course, you can find on eBay sellers of non-compliant and even dangerous sockets that are aimed at the UK market but which do not meet UK standards and which would be illegal to sell in the UK.
The specifications for typical UK sockets does not allow for mixing wires of different sizes.
Source: MK
As you can see from the example spec above, the terminals are designed for clamping multiple wires together.
A normal 32A ring main uses 2.5 mm² conductors, you can clamp three per terminal (e.g. in, out, spur)
So long as you check the screw is tightened fully and then check that you cannot pull any wires out, it will be safe.
I believe the intent here is that a gas-tight connection be formed between the conductors and the clamp. Copper is a fairly malleable metal so it should normally be distorted by the clamping mechanism such that the surfaces formed on the wires conform closely to each other and to the clamping surfaces.
As you note, its a loose connection that causes overheating and burning.
It isn't normal to see any significant corrosion other than normal superficial tarnishing of copper from exposure to air.
Why don't sockets just have two pairs of screw terminals that are connected internally?
You can get sockets with dual terminals, but these are specialist unswitched sockets for use in medical applications - each of the terminals can still be used to clamp multiple wires (four 2.5mm² in this example)
Source: M2
Dual terminals for live, neutral and earth Each terminal accepts 4 x 2.5mm²
Screw cap covers minimise dust & dirt traps improve effectiveness of infection control procedures
Clearly engraved 'MEDICAL EQUIPMENT ONLY'
Face plate manufactured from a blue urea compound which contains a very effective life-long anti-microbial & anti-fungal additive
Isolated Power Supply sockets for medical applications
In line with HTM06-01 and MEIGaN requirements with clean and dirty earth connections
These would be inappropriate for use in a home.
Notes
¹ If we set aside the issues with split ring-mains and treading on BS1362 plugs.
Best Answer
The earth connection should be a ring terminal under the screw. With the earth connection it will be safe to touch the ballast case in all circumstances
Dry wood does not conduct electricity, nor does it pose a fire threat due to electrical faults
I'm not a big fan of those pliers crimp connectors. (but that's personal opinion)
The ballast will be safe and come to no harm with no lamp connected.
Some sort of strain-relief clamp should be arranged on the jacket of the appliance cord so that yanking the cable does not disturb the connections.
I would not expect the ballast to get hot get hot enough to start a fire