It all depends on how important not clobbering things when someone makes a mistake, or when SNAFU happens, is. I'm going to assume that's important.
For example, you give an example in foo:
throw new exception("Method should not be called if D==F, you forgot how to use this method");
Assuming that foo doesn't set D and F, and the programmer does, and of course that foo isn't supposed to handle the situation where D and F are equal, it's probably the right choice to throw an error. Programmers are idiots, especially the one who uses your code next.
As for checking that something was successful, that's not a bad idea if you are able to identify possible points of failure. For example, assuming a HTTP grabber that returns a (integer flag, data object) tuple for whatever reason:
(flag, page_data) = http_fetch(bleh);
if (flag not in ACCEPTABLE_FLAGS) {
throw new exception("Today is not a good day.");
}
file_thing.write(page_data.bytes);
Here, the error checking will mean that nobody knocks down your door because the internet went down and as a result your code clobbered the databases with random junk.
If it's your own work, e.g., that you wrote http_fetch above - do the statements anyway. Unless you're in a really terrible, terrible work environment, people should appreciate that you are just recognising that sometimes the worst case scenario happens.
On the other hand, it's probably excessive to do the following:
mid = (low + high)/2;
assert((low <= mid) && (mid <= high));
In short, if you can identify a point of failure, cover it. Even if it's your own code, sometimes things go wrong and the assumptions you made when writing that code - such as that the caller would actually read the docs - are incorrect.