Let me start by saying that what I am about to write are just observations on gameplay and what I've seen in the code. I am not positive that this is exactly how the game behaves, but I think it's pretty close.
War Weariness is not FfH-specific. It is an absolute number (like the absolute Armageddon counter.) The only time it increases is when you lose a unit to a civ you are at war with. I think the amount the weariness counter increases has to do with the relative strength of the unit lost. This weariness number is then divided by an arbitrary war tolerance number and applied to all of your cities to create a % of weariness.
There are civics and buildings that adjust your weariness. If your civ has the City States civic, for example, you will have a +25% war

in each city. Each weariness point counts 25% more. For each city that has a dungeon (-25% war

) this penalty will be negated.
The weariness % becomes an unhappy citizen (rounded down) under the listing of "War, what is it good for?". For example, if you have 10% war weariness in a size 9 city you will have no unhappiness. Once the city increases to size 10, it will have one unhappy citizen. If the city size becomes 20, or the war weariness becomes 20% then you'll have two unhappy citizens.
If you are at war with multiple civs, all of the weariness is added together to determine unhappy citizens. Weariness only comes into play when you are at war. If you sign a peace treaty you will not suffer happiness penalties (nor religious war penalties) and your weariness will decrease slowly over time.
The "We will not fight with our brothers..." happiness modifier is completely different than the "War..." modifier. I believe it has to do with the % of people in a city that follow a certain religion. If you are at war with a civ that follows a certain religion, a certain % of all of your citizens who follow that religion will become unhappy. Each of your cities that has this religion in it will be effected depending on how many citizens in the city worship under that religion. I think there is a leader-specific religious tolerance factor that comes into play as I see this penalty more often with some leaders than others.