You're welcome. Yes, the dot is the pin 1 marker.
Yes, the internal block diagram, as you have shown is what you need, you can see that it is split into two 'halves, the only shared connections are supply and ground. They look a little squashed in the 556 due to the pinout
*. I have attached an equivalent block diagram of one section from a TI 555 datasheet. This is drawn a bit more helpfully (inputs on the left and outputs on the right) which makes it easier to visualise and hang the external passive components on. I've deleted the 555 pin numbers so you can copy and add the 556 pin numbers for the appropriate half.
The way the meter translates is by integrating the width of the pulses. If the pulses are very narrow, then the average voltage is near zero so the deflection is very small, if the pulses are wider then the average voltage is higher, and so the meter deflection greater. The way to calibrate the meter would be to find a fairly accurate (eg 5% film) capacitor of 10nF, 100nF or 1uF to set full scale on one of the upper three ranges (as there is one pot, the appropriate capacitor should then show full scale on the other 2 ranges, within reasonable limits). For the bottom range, you will need a 1nF capacitor to set the other pot. You can then experiment with other capacitors, eg 22nF/220nF and 47nF/470nF, to see how linear the meter reading is. Off-hand, I'm not sure how linear the meter response will be (worst case, you might need to create a graph of meter deflection vs capacitance).
Yes, an ESR (Equivalent Series Resistance) meter uses an AC signal (often squarewave or 'modified sine wave'), normally around 100kHz (to cater for SMPS capacitors). These treat the capacitor as a 'shunt resistor' from the AC signal to ground. The magnitude of the signal that gets past the capacitor represents its ESR. There is a popular thread around about making a good workable ESR meter with just a 5 transistor circuit. It would be worth searching this out if you want to build one. [Edit:
https://www.eevblog.com/forum/projects/5-transistor-esr-meter-design/ ]
EDIT:
* While attaching the 555 block diagram I noticed that your 556 one doesn't show the internal voltage divider resistors on the inputs of the comparators (they
are in there). This voltage divider resistor chain is important to understanding how the Threshold and Trigger inputs and comparators work!