I know that a character's strength can never go below zero, but what about other numeric values in the game? For example, both players reveal a 0 initiative plot, but player A gets a -1 modifier from Burned and Pillaged. Do we decide who wins initiative by total power, or does player B win it?

Second example, a bit more extreme: A player reveals a low cost plot while having enough negative modifiers to get the income below zero, while he still has some gold left over from Planning Ahead. Does he actually have to pay gold? I'd imagine not, and it wouldn't make much sense to have claim or cost drop below zero either, but I can't find anything in the rules to stop them, or explain what happens.

Keep in mind that it's not so much that a character's STR can "never" go below 0 as it is the effective STR is rounded up to 0 if the result of the STR check is less than 0. This is very important for subsequent modifiers. For example, say you have a 1-STR character with Poison Wine on it. Mathematically, 1 - 2 = -1. But since we round up, the effective STR is 0. Now, that same character gets +1 STR (say, Knighted or Dragonbone Dagger). It's new STR is not 1 (0 + 1 = 1), but 0 (1 - 2 + 1 = 0). So there's a difference between the character's STR being 0 and that STR being rounded up.

As for other calculations, there is nothing specifying that the results be rounded to 0 if they come out negative. However, what you do with the negative value is going to depend on what the rules tell you the calculation is for. In short, is there a consequence to something coming out negative? Looking at your two examples:

1. When figuring out who wins initiative, you give it to the person whose count is highest. Well, 0 is higher than -1 and B wins initiative in your example. The negative count doesn't affect what you do with the initiative count.

2. As for income, it might be possible to count an income that is less than 0, but since there is nothing in the rules saying that you have to "pay back" a deficit in income count, there is no consequence for doing so. Don't make one up. The player would keep any gold that happens to remain in his/her gold pool.

Page 23 of the Core Set Rules seems to indicate that rounding numbers up to zero is exclusive to character strength:

"If a character’s STR is ever lower than 0 after all effects are applied, its STR is rounded up to 0. Any time a new modifier is added to the mix, reevaluate to net sum from scratch before applying it to the base STR."