For an inline 4 cylinder, there are two combinations that can be run, those being 1,2,4,3 and 1,3,4,2. The reason this is the case has nothing to do with the camshaft but everything to do with balancing the crank as it spins (crank end balance).
The two outer pistons are weights that fly up and down together. Likewise the two inner pistons do the same. What happens here is that when you look at the crank (picturing all 4 pistons going up and down), you have effectly (lets say almost - but not entirely) cancelled the crank end forces. This way, the crank doesn't want to pull-up or pull-down excessively on the end journals due to having a different crank journal arrangement. So the crank is machined with #1 and #4 journals & #2 and #3 jounals 180 deg to each other, purely for reciprocating weight balance reasons.
So accepting that its all due to crank end balance, the two outer pistons must coincide with each other, just as the two inner pistons must as well. See picture below (top picture is a balanced crank, bottom picture is an example of an unbalanced crank that i whipped up & would never exist in the real world)
Given this, then the camshaft must be machined to match one of the two applicable firing orders. Once the manufacturer has settled on one, then thats what it becomes. You are correct, that its the camshaft that dictates the firing order, but this is still a by product of solving crank end balance.
To animate the firing in your mind:
1243 does this |-->|<--|
1342 does this |<--|-->|
The sohc can be run from a distributer (just like a car) and some have done so in the past. The reason waste spark is employed is simply due to economics. Its more compact and cheaper to just fire on both the power stoke as well as the end of the exhaust stroke in an attempt to do away with a distributer.
It is quite possible that a motor could run in reverse, but your camshaft also needs to be reversed. The motor is just a air pump, so it wouldn't care providing the valve timing and ignition timing is correct in the reverse direction. Of course, then you need to start looking at reversing the pistons as well because they have an arrow pointing to the front for a reason (because the gudeon pin is offsetted). And on some newer motors (more so these days), the actual block is offeted as well to both increase torque and reduce piston slap. Not to mention that your oil pump is now sucking instead of blowing. So you'd be ending up with a case of all sorts.
Interesting question, its been thought of before. The Ford Kent motors are 1243, so its not that cars don't use that firing order. Lots of cars use 1342.
So to sum up, yes, if you machined up a camshaft to fire with 1342 instead of the 1243 that we have, your sohc would run just fine. You wouldn't even need to touch the ignition timing. Let me know how you went
