Stranded
Flexibility
Stranded wire is more flexible than solid core wire, which makes it well suited in situations where there will be movement of the wire. It also makes routing the wire through a building a bit easier.
Higher resistance to metal fatigue
Where solid core wire will weaken and break, stranded wire holds up to being bent repeatedly.
Solid
Cheaper to manufacture
While this savings may or may not be passed on to the consumer, solid core wire is cheaper to make.
More rugged
Solid wire resists cutting and stretching damage better than stranded wire.
Lower resistance
Given a solid and stranded wire of the same diameter, the solid wire will have a lower resistance. This is because there will always be gaps between the strands of a stranded wire. A stranded wire of the same gauge, will have a larger diameter than a solid wire to compensate for this. When you buy wire, you'll always buy it based on gauge. Both solid and stranded wires will have the same resistance, the stranded will simply be a bit larger in diameter.
Easier to terminate
If you're not careful when stripping stranded wire, you can end up cutting off strands (usually only a problem with smaller gauge wire, or wire with lots of strands). Depending on the number of removed strands, the ampacity of the wire can be negatively affected. Though nicking a solid core wire while stripping can make the wire more susceptible to breakage, so this isn't a clear cut advantage.
Solid core wire can be used in push in ("backstab") terminals, more often than stranded. This can be a pro or a con, depending on who you talk to.
Stranded wire can be more difficult to terminate on screw terminals, as you can sometimes have escapee strands which don't end up under the screw. This again can affect the ampacity of the wire.
Because stranded wire has a larger diameter, it tends to fill up twist-on wire connectors faster (less wires allowed in the connector).
Unless there's a lot of movement within the home (RV, house boat, frequent earthquakes), or you're pulling the wire through conduit. I'd say there isn't much of a difference either way.
This is OK provided that the wires to the fan control switch are treated as 120VAC wires as per 725.46:
Class 1 Circuit Wiring Methods. Class I circuits shall be installed in
accordance with Part I of Article 300 and with the wiring methods from the
appropriate articles in Chapter 3.
(i.e. run using NM and not communications-type cable/wire) and as per 725.48(B)(1):
(B) Class 1 Circuits with Power-Supply Circuits. Class 1 circuits shall be
permitted to be installed with power-supply conductors as specified in
725.48(B)(1) through (B)(4).
(1) In a Cable, Enclosure, or Raceway. Class 1 circuits and power-supply
circuits shall be permitted to occupy the same cable, enclosure, or raceway only
where the equipment powered is functionally associated.
since the control wires are with the fan's power wires, and we can safely treat them as a Class 1 control circuit using the provisions of 725.130, Exception 2:
Exception No.2: Class 2 and Class 3 circuits shall be permitted to be
reclassified and installed as Class 1 circuits if the Class 2 and Class 3
markings required in 725.124 are eliminated and the entire circuit is
installed using the wiring methods and materials in accordance with Part II,
Class 1 circuits.
Best Answer
If your appliance requires a voltage greater or lesser than you have in your household electrical system, then you must supply the required voltage. Electronic devices should have a label or embossing with their input voltage, and many like phone or laptop chargers might accept a range of 100 to 250 volts, meaning they can be safely used in both the US and the UK, for example.
I personally moved from the UK to Mexico and have rewired an extension lead so it has a North American plug and British socket outlets, so I didn't have to wire new plugs onto my Blu-Ray player and Apple TV. I don't have to worry about voltage since the supply in Mexico is within the range accepted by those devices.
If you buy a bedside lamp, you'll need to change the bulb, but the rest of the object is just some cable, a switch, and a structure. As long as you're not putting an extremely high power bulb in it that would exceed the rating of its internal cables, you've nothing to worry about, and I would simply go to Home Depot and buy a plug to wire onto it. If you're not up to that, buy a simple but robust travel adapter. This will serve for long term use.
Now you mentioned a coffee maker, my Bialetti was brought over from Italy and I rewired it with a UK plug, but according to the appliance label it requires 220V+ so for use in Mexico I need a transformer to step the voltage up from 110V to 220V (approx., this is Mexico). You have to take into account the wattage (power rating) of the device so you get a suitable transformer.
Frequency (50Hz in the UK, 60Hz in North America) can be a consideration for some devices, as manassehkatz pointed out in a comment, and this would require additional equipment the cost of which would be prohibitive for household appliances.
My recommendation is that a 110V to 220V transformer of sufficient power rating, costing perhaps $100 or less, would be suitable for long-term use with a European appliance that requires 220V, while any device accepting 110V should be preferably be rewired to use a US plug, or, failing that, be connected to an individual non-transformer travel adapter that makes a good fit with both the device plug and the US outlet, or like I do via a foreign power strip wired to a US plug. It is even possible to purchase pre-made strips in this configuration.