NOTE: Not looking for real-life medical advice, though the question is inspired by me going through a weird two-week coughing spell last month.
Easiest example would be influenza. If you go to a doctor with several flu symptons, but absolutely no fever at all, will a diagnosis of “influenza” be dismissed out of hand? Would the absence of a fever mess up various kinds of infection diagnoses (i.e. “Nope, you can’t have Condition X because you have no fever … but wait! Your labs show wholesale infection! 
”)?
Are there cases in the medical literature of folks being physiologically unable to muster up a fever in response to infection? If so, I assume that is a dangerous situation to be in, yes?
Last question: body temperature varies among healthy individuals as I understand it – 98.6 is “merely” an average, correct? If so, do the upper-end body-tempeature outliers get any kind of infection-fighting advantages from basically having a permanent low-grade fever? Do “never get sick” people typically walk around with body-temps of 99.2, none the wiser?
A true stumper … that’s pretty rare around here 
Typhoid Mary never had a fever. About 5% of typhoid victims have no symptoms, but still spread the disease and need to be treated.
Yes, and it’s not at all rare.
Fever is often absent in the elderly even when they have an illness usually associated with a fever. More generally, any state where someone is “sick” or immunocompromised may lead to that person not developing a fever even during illnesses where fever is common. Specific examples are people with cancer, people with diabetes, and people taking prednisone or similar drugs (e.g. hydrocortisone, dexamethasone, etc.)
It should also be noted that even the healthiest individual may fail to mount a fever if they have an overwhelming infection. In that case, the absence of a fever portends a poor prognosis.
Regarding H1N1 specifically, fever is absent in about 10 percent of victims.
While fever is one of the signs we look for, it is neither pathognomonic for infection nor does the height of the fever correlate with how ill the patient is.
A persistent cough can have many non-infectious causes as well as infectious ones, and fever would not be a particularly helpful or reliable sign in distinguishing any of them although on average an infectious cause is more likely to produce a cough with a fever than is a non-infectious cause.
I hardly ever get a fever, despite getting my share of colds and other minor infections. My general health is good, I’m not on any medication that might affect my ability to mount a fever response. As far as I know, that’s how I’ve always been - my parents learned the hard way that absence of fever didn’t necessarily mean I wasn’t ill.
When I have a cold or flu, I’ll often feel feverish, but my temperature hardly ever goes above 36.5C. I don’t think I get sick any more or less than average.