It's not a standard part. Most transformers have mains voltage primary (typically 110V or 230V) not 24V.
Yes, it should be easy to measure them with a multimeter: the secondary with the highest current rating should have the lowest DC resistance.
I agree, it's confusing. The transformer appears to have four windings: a 24V primary and three 15V secondary windings, two rated to 15mA and one rated to 40mA.
No you can't achieve a very high current. Regardless of the capacitors used, the 555's maximum output current is 200mA, which limits the maximum current from the circuit to 100mA, at which point there will be a large voltage drop.
Without a part number or datasheet there's no way of knowing.
Generally infra red LEDs have a higher current rating than visible LEDs because their forward voltage is lower (1.1V to 2.5V, as opposed to 1.8V to 4.5V) and the maximum rated power dissipation is similar.
Don't worry, it's normal for a power supply to have a slightly higher voltage when unloaded. If you connected the laptop and turn it on, you'll probably find the voltage drops to within 19V or so.
It seems like an overly complicated way of doing this.
I think you should drop the MUX and three channel photodiode, use a single channel photodiode and turn the LEDs on, one at a time and take an intensity measurement for each colour.