Popping from the exhaust usually comes from either incorrect pressures, or lack of exhaust valve sealing. Wrong pressures can imply things like the wrong camshaft timing, the wrong ignition timing, abnormally slow combustion, or high back-pressure in the exhaust port.
If the engine starts properly and runs properly except for the odd pop here and there, the main things to focus on are poor valve sealing, or high back pressure. High back pressure would come from something like a clogged muffler. When the muffler is clogged the power is way down, and the exhaust valve will have been very hot. Overheated exhaust valves have no carbon on them, and are a distinctive flat grey-black colour.
Before we got as far as looking for some of those issues, you reported fairly substantially wrong tappet clearances. It can't be expected to run properly like that, tight tappets are very likely to cause leaky valves, and it isn't difficult to fix the tight tappets, so the first step is to fix them. If the problem had persisted after that fix, it would have been time to remove the muffler, inspect it, and run the engine without it to see if the problem went away. If it hadn't been the muffler, we would have continued down the list of possible causes until we found it. Meanwhile we'd have been making it a better engine.
So, a trade secret: sometimes I'm not actually sure the first thing we look at will turn out to be the problem. I just try to start with the things that are both the most likely to be the issue, and the easiest to fix. Most car mechanics did the same thing, in the days before OBD (on-board diagnostics, a feature of all modern powertrain computers) and plug-in electronic diagnostic testers that read the OBD records and report the fault conditions that have been recorded while the car was being driven by its owner.
