Electronic – How to find a short in an unpopulated 4-layer PCB

I Think you are using the term plating incorrectly. Plating will decrease the diameter of the hole, not increase it.

The larger dimension (the pad surrounding the via hole) is called the Annular Ring.
All the fabs I have worked with generally want a 0.005" per-side annular ring, or the via diameter + 0.010".

You definitely have some really hairy registration issues on the board you posted pics of. It may work, but you're really pushing it.

Generally, you never want the via hole to break out through the edge of the annular ring, which is happening a few times in the picture you posted.

Anyways, Registration refers to the accuracy between a fab-house's etching and drilling process.
Basically, if a fab house etches a circle in the coper of a board, and then drills a hole in the middle of the circle, how close is the drilled hole to the center of the copper circle?

Remember, drilling a board and etching it are separate processes, and involve the board being unmounted and remounted in different equipment.
Generally, you can get as good registration as you are willing to pay for, and it looks like your boards are from a pretty budget board-house.

You need to allow enough annular ring that you never wind up with the via hole too close to the edge of the pad. This is generally specified by the fab house (they should have a minimum annular ring spec on their board requirements). However, it is important to remember that they may run your boards anyways, even if it does not meet their minimum required specs.

The board house may run the boards anyways, and just refuse to fix any issues if they don't work out.
This is particularly common in China, where the general philosophy seems to be "Let the Buyer Beware".

Anyways, I think the reason you are finding your vias a bit odd looking is that you have tented your vias, which is the practice of covering the annular ring and the hole for the via with soldermask.

With 0.015" vias, you will occasionally get a contiguous layer of soldermask over the hole, and they will look different.

I've decided that breadboarding ahead of time is a waste of time unless there are aspects of a part's performance that you need to measure yourself.

I even think building the board in pieces is a waste of time. The initial hardware debug just takes a day or two. Any subtle issues won't show up until software integration anyway. I'd rather debug one circuit (the complete one) than several ad-hoc reductions of the circuit.

Just build the thing! Assemble the whole board and power it up. Look at the supply voltage. How many mA does it draw? Which part is burning hot? Which part is warm? Then try to burn firmware on to whatever microcontroller is on the board. Then start writing firmware. Get the clock going and toggle a pin. Get serial (or whatever) communication up. Now write test programs for each peripheral. Then build the production test fixture and start writing the "real" firmware.

Edit:

On finding stuck rails

If a supply rail is stuck to ground, (measures 0 ohms with an ohmmeter), power it up on a bench supply. Set the voltage normally and the current limit to a few hundred mA. Print out the PCB design on paper, and find a DMM that measures microvolts. Measure microvolts starting at the supply terminals, and write down the voltage drops on the PCB printout. By looking at microvolt differences, you can trace exactly where the current is going without depopulating or chopping up the PCB. This technique is a better than using an ohmmeter because you can pump a lot of current through the circuit, more than any normal ohmmeter will supply.

A similar technique works for finding shorted traces on an otherwise populated board. Run the board a use a scope to find "digital" traces that are in an in-between voltage range.