1) What does "burn the fuse" mean in "I'd totally burned the fuse thinking about it"?
This is a guess: It probably means that I'd run out of ideas, or run out of energy.
2) What does "slant" mean in "It was just in contrast to the reporter's slant"?
"slant" means bias. It is suggesting that the reporter incorporated some of their own ideas and beliefs into the way they reported the story.
3) What does "streak" mean in "You didn't get that stubborn streak from me"?
A 'streak' is a personal characteristic. It is often used negatively. "He sometimes loses control, and you see his violent streak"
4) What does "sizzle on" mean in "I'm not trusting myself with the words sizzling on my tongue"?
Again, this is a guess. I think it may mean that they want to say something, but are restraining themselves.
5) What does "hold out" mean in "I do hold out hope for the boy"?
"hold out" here is something like "still have". It has the general idea of "persevere". The idea here is that you still do it, despite the fact that what you want is unlikely to happen. "They don't have any food or warm clothing. We hope that they can hold out the night until the rescuers can come in the morning".