Electronic – How much power (in watts) can I get from the house outlet

The Power over Ethernet (PoE) is likely most suitable if the devices are network-aware, otherwise one Ethernet cable is pretty much equivalent to a (low current) power cord or a USB cable.

For any devices that to be "worn" or "attached" by test subjects, I would strongly consider an un-tethered design, powered using rechargeable batteries. For small signal (i.e. sensors, data logging, no motors, very simple LED lighting) this option should be easily economic and practical for up to 24 hour periods or longer. NiMH would be my first choice due to low cost and wide availability, with Lithium as a second. Just ensure you use an appropriate charger for the battery type, and things should just work.

Note that most rechargeable batteries such as 'AA' size cells, only provide 1.2 rather than 1.5 volts, so 4 of them is not sufficient for 5V needed to stably power an Arduino, while non-rechargeable (i.e. disposable) cells such as alkaline or zinc-carbon would.

With electricity you need to be aware of both voltage and current. The Arduino itself takes an input range of 7-12 DC, up to 500mA (I don't have a reference on average / typically current, but would guess around 100-125mA or less), from an external DC power source. The USB port can also be used to draw 5 volts, up to 500mA from a powered USB hub or powered USB port.

Using a low-cost low current (e.g. 250-500 mA output) AC-DC power adapter (wall wart) would be a default method if there is non-trivial power consumption, or needs to continuously operate for a long period of time. A modern switch mode power supply (SMPS) based unit can be had for a modest cost, and is light weight, being able to dispense with the need for a large power transformer encased in it. Combined with the power limiting capabilities already included in the Aruino (UNO) from the resettable fuse (PTC, I believe) used for USB power source, and/or the linear voltage regulator used to regulate power from a AC-DC power adapter which also includes over-voltage, current limiting, and short circuit protection unless you need addition power requirements (e.g. motors, high power LEDs) you can use the protection built into the Arduino as sufficient for electrical shock/burn protection.

I can help to address the first part of your question. As an abstraction of a single switching device inside a processor, imagine a MOSFET connected to ground with a load resistor to the power supply (in a real CMOS part there wouldn't be a load resistor, but another transistor, but this distinction is not important for the analysis). Connected from the junction of the resistor and the transistor is a capacitor, representing all the input capacitances of the transistor that the transistor under discussion is driving. When the first transistor switches off, this capacitance will be charged up through the load resistor. When the first transistor switches back on, the charge stored on the capacitor will be discharged through the first transistor.

It can be shown that when a capacitor is charged through a load resistor, 1/2 of the energy used in charging the capacitor is lost in the resistor, for a total energy dissipation of \$\frac{1}{2}C{V_s}^2\$, where \$V_s\$ is the supply voltage. When the switch then turns on, assuming the resistance of the switch is much less than the load resistance, that same energy will be dissipated in the switch, for a total energy of \$C{V_s}^2\$. Dividing this by the switching period gives you the dynamic power dissipation of the switch/capacitance combo, \$C{V_s}^2f\$. Shrinking the die reduces the junction capacitances of the MOSFETS, so if you know the supply voltage, switching frequency, number of transistors and the approximate junction capacitances of a certain process, you could calculate a ballpark figure of what kind of power savings a process shrink entails, all other things being equal.