Favored class options and fractional bonuses
I know it's odd, but let me directly address this part of your own answer:
Dyndrilliac: If fractional bonuses were intended to be rounded down prior to being added together, then logically, the FCO would never provide any bonus because it adds +1/2 each time it is selected. So in order to receive any bonus at all, it must be selected multiple times and those individual +1/2 bonuses need to be added together prior to rounding in order to ever result in a positive integer.
Emphasis mine. I get what you're saying, and I understand your frustration. I fear, however, there's an intermediate step that might've been overlooked. Racial Favored Class Options says
Many... alternate class rewards add only 1/2, 1/3, 1/4, or 1/6 to a roll (rather than 1) each time the reward is selected, or add 1/2, 1/3, 1/4, or 1/6 to a class ability.... When applying such a benefit to a die roll or a class ability, always round down (minimum 0). You might thus need to select such an option several times before the benefit applies.
And Alternate Racial Rules says
When choosing one of these favored class options, the reward is gained on a level-by-level basis. Unless otherwise noted, these benefits always stack with themselves.
Emphasis mine. Hence, a favored class option bonus stacks with itself even if it's a fraction. That is, a favored class option is picked, the favored class option bonus is computed and stacks with itself, and then the rounding happens. For example, a favored class option that's +½ and picked twice isn't two +0s (+½ each time and rounded down each time) but one lone +1. Favored class options are a rare case of specifically stacking fractions, and keeping fractions doesn't happen with, for example, the inquisitor's extraordinary ability stern gaze, which just generates a value that can be a fraction and is rounded down when it does.
A fractional bonus from a favored class option does not get rounded before it's added with itself for the same reason that a creature picking the generic favored class option +1 hp twice means the creature gets from that option +2 hp instead of only +1 nonstacking hp twice: it's the same source so it normally doesn't stack, but the game specifically says it does so it does but only with itself. But the special ability stern gaze has no provision for keeping fractions with itself or another feature, so the special ability stern gaze is always rounded down normally before it's stacked with anything; there's just no option to add +½ of a bonus from the special ability stern gaze to another +½ from somewhere else as stern gaze is separately and independently computed as "1/2 [the inquisitor's] inquisitor level (minimum +1)."
I'm unaware of any other sources that stack fractional bonuses before rounding. Unless the argument is that The game does not say you can't stack then round, there's no way for a half-orc or tiefling inquisitor that takes the inquisitor favored class option to do so. That means, unfortunately, the way it works is the way that folks that disagree with you say it works.
On rounding fractional bonuses
The bonuses from the the inquisitor's extraordinary ability stern gaze and the half-orc or tiefling's inquisitor favored class option are, obviously, from different and discrete sources. The bonuses can't be applied before rounding because each bonus already exists separately as a unique bonus.
That is, the first feature grants a +1½ bonus therefore granting the creature a +1 bonus (if nothing indicates otherwise), and the second feature grants a +1½ bonus grants therefore granting the creature a +1 bonus (if nothing indicates otherwise), and, because those bonuses already exist, it doesn't matter to the game that those bonuses end up applied to the same third thing.
A house rule can fix this by, for instance, keeping fractions when two or more things add fractional bonuses to the same thing, but that house rule's contrary to—I'm pretty sure—everything else in the game about computing bonuses, requiring confirmation first that another fractional bonus will be added to that third thing before knowing to keep the fractional bonus.
Really, when determining a bonus's value, the game usually cares less about where a bonus goes (seeing after it's computed if it stacks, for example) and much more where it's from (because that usually determines its value), so such a house rule may create some confusion, especially among long-time players.
