Best method/circuit for converting 240VAC to 5VDC (Low power)

You can't rely on using resistance measurements to get turn ratios, because they almost always use different gauge wire for the two windings. The primary winding is usually several hundred turns of some fine gauge stuff, while the secondary is a few dozen turns of something heavier, to carry the necessary current.

Probably the easiest solution is to get a 2nd wall-wart that supplies a few hundred mA of regulated 5V, and you're done.

However, if you really want to DIY it, you may be able to find transformers with 24V secondaries that are center-tapped, in which case you may be able to use one leg to the c.t. to get 12V AC for your diode-and-7805 approach.

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.