In sentences 1-4 you have integrated relatives (often called "restrictive"), the sort that isn't surrounded by commas. You can tell these are relative clauses because they have gaps—specifically, their subjects are missing.
The gap in each case is left behind by the relative pronoun which. But in your examples, you've replaced these relative pronouns with that. You can undo this change and put which back in its rightful spot, if you like:
1. The darts contained a drug that/which [ ____ put the wolves to sleep ] .
2. Few kids in the survey met the guidelines for physical activity that/which [ ____ raises the heart rate and makes you breathe harder ] .
3. Pi came to an island that/which [ ____ is sweet by day but dangerous by night ] .
4. Other major effects have included programs to destroy rabbit holes or use viruses that/which [ ____ sicken and kill rabbits ] .
If these were supplementary relatives (often called "non-restrictive"), the sort that's usually surrounded by commas, you wouldn't be able to replace which with that; you'd be stuck with which in each case.
In sentence 5 you have something very different! The clause following that is complete and therefore contains no gap. And without a gap, there's no place for a relative pronoun like which. The word that, therefore, cannot be taking the place of which; there's no substitution to undo, because which wasn't there in the first place.
Instead, this particular that is a subordinator, introducing a subordinate finite clause:
5. The answer is probably as simple as the fact that [ people change ] .
Since this isn't a relative clause, the relative pronoun which has no place here. Since which isn't an option, you're stuck with that in this example.
No, using would not fit.
This is the noun use (pronounced with an /s/ sound, not a /z/). That has one meaning which is something like the act of using, but here it has another meaning which is approximately usefulness.
In fact, of use is an idiom which means the same as useful.
So if you saw
Use of this equipment is prohibited.
that would be the first meaning, and so Using this equipment would be equally good.
But in phrases such as of use, any use, no use, it always has the second meaning: you could substitute usefulness, but not using.
Best Answer
When used the way "odds" is used in your question the two are pretty much interchangeable, but odds are usually presented as a set of numbers related by to, while chances are a set of numbers related by in, and the numbers mean different things.
When chances are presented using in, the numbers convey the number of chances out of a total. For example, 1 out of 4 means that out of 4 tries you are likely to win once.
When presenting odds using to, the numbers say how many time you're likely to win to how many times you're like to lose. For example, odds of 1 to 4 means that you are likely to win once and lose 4 times. so when the odds are 1 to 4 the chances of winning is 1 in 5.
When the chances of winning are 1 in 1,000,000 the odds are 1 to 999,999 but the difference is so small as to be all but negligible. So in things like lottery odds the terms are pretty much interchangeable.
The key point to notice is whether they use to or in. When they use in they mean out of, that is, the second numbers represents the total and not the number of losses regardless of which word (odds or chances) they use.
Put another way: To should never be used with chances
but, in can be used with either