I will start with the last question:
The power adapter is a very cheap one. In fact, the voltage without load can raise up to 15-16V. 9V is the nominal supply voltage under the normal load that the Spectrum circuit imposes.
Why 9V? It was a "standard" voltage setting at the time (equivalent to six 1.5V batteries), so transformers whose secondary winding generated a 9V AC power signal were cheap and easily available for the manufacturer. The Spectrum circuit needs about 1A, so anything higher than 9V for that amperage would have required a bulkier, hevier (and possibly more expensive) transformer. Remember that Sinclair was aiming for a cheap product.
Why not 5V? To start, 5V was not a standard choice for external supply (because of the above reasons). In the case of the Spectrum in particular, 5V would have been too far from 12V, which is needed for some devices inside the computer board.
The ZX Spectrum components have different needs regarding the supply.
First, the logic circuits: these operate by standard TTL/MOS levels, they are the most greedy of all them, and the supply must be very precise. TTL circuits use 5V, so a standard 7805 regulator was used. This is a linear regulator which accepts an input voltage from 7 to nearly 30V, and outputs a steady and virtually noise-free 5V signal. Linear means that the excedent of supply is disipated in form of heat. For a 9V DC input and 1A, this means: (9-5)V * 1A = 4W of heat disipation. Using a higher voltage, such as 12V, for input, would have increased the heat that the regulator would need to disipate. The 5V supply rail feeds the CPU, ULA, glue logic, ROM, and non contended RAM memory (the memory that goes from 8000h to FFFFh in the Z80 address space). It also feeds the emitter-follower circuit that drives the loudspeaker, although in some board issues, the emitter follower is connected to 9V instead of 5V.
The video encoder chip, LM1889, uses a supply voltage of 11-13V. Contended RAM memory (the one that is addressed from 4000h to 7FFFh) also uses it, so a way of generating 12V out from 9V or 5V is needed.
Contended RAM memory also uses -5V, so this must be generated as well.
The method is to use an internal DC-DC boost converter. It's depicted here:
NOTE: signal labelled as 5V (20B) is actually -5V (20B is the pin at the rear connector where this signal is available)
It is esentially an inverter circuit. TR5, TR4 and the coil forms the inverter. It generates a pulsating 12V signal at the collector pin of TR4 (it comes from the reactance discharging through transistor TR4). That signal is AC coupled by C80, so we have something similar to a 12V AC power supply, of about 110kHz, at the right side of C80.
D15, D17 and C44 form a simple rectifier and a simple filter. As frequency is much higher than 50Hz, no full rectifier or big capacitors are needed in order to have a steady power supply. The junction at D15 and C44 carries a 12V DC signal suitable for DRAM chips.
The same voltage is passed through a PI filter formed by C44, R62 and C45 to give another 12V supply rail, isolated from the first one. This isolation (achieved in some way because of the PI filter) is needed to avoid high frequency switching noise coming from DRAM chips to enter the video circuit section, which would make it to display on screen, degrading picture quality.
To generate -5V, a similar circuit is used: from the same point the 12V supply is got, another capacitor, C46, gives a 12V AC signal. Diodes D11, D18 and capacitor C79 play a similar role to D15, D17 and C44. Only that this time, diodes are arranged in the opposite direction, so the output signal is negative (with respect to ground). D19 is a 5.1 zener diode, which is used to lower the voltage from -12V to -5V. In order to not to destroy the zener, a current limiting resistor R79 is used. The -5V supply is very weak and only used to polarize the substrate of DRAM chips.
This particular DC-DC converter is the cause of many broken Spectrums. It happens that 4116 DRAMs (the ones that need three different voltages and forms the contended memory block) require them to be applied in a very specific order. Order that the DC-DC circuit doesn't respect. If this happens, there is a chance that the DRAM memory is damaged. Without the contended memory block in working order, the machine cannot boot and the ROM hangs the computer. The user doesn't see the (C)1982... welcome message, but a screen with light gray, white and/or black vertical thick stripes. The pattern may vary. The border is always white (light gray actually).
Additional information can be found at section 6 (power supplies) of the ZX Spectrum 48K Service Manual