Okay, I'll give this a shot as there is misinformation on misinformation correcting misinformation with some misunderstandings...
First, 1,250 is correct. Duty time (not flight time or credit time) is used. As of right now, the only relevant case that I am aware of to come before a judge is Knapp v. America West in which it was ruled that only Duty time counts. Knapp argued that time spent on reserve should count toward the 1,250. The court ruled that reserve time DOES NOT count. The court left open the fact that TAFB may count, however, it did not rule on this fact because it was not necessary to do so in this particular case (Knapp would not have had enough hours EVEN IF TAFB was counted.) The court also left open the possibility that the frequency of times called while on reserve may be relevant in a future case, but was not ruled on in this case as Knapp did not raise this argument.
As the law is written, many if not most line holders should qualify as the law only requires 104 hours per month.
The airlines count on the fact that we will go into HR and take there word for it. At most we will call the DOL and get told that the airline is doing what they are doing "in accordance with the law" and then stop. What they don't tell you is that a lot of the law is unsettled. For example, the court in Knapp vs. America West alluded...but did not rule...that time spent on overnights should be counted toward the 1250. If it went to court again with a different case, I believe that there is a very good chance that a court would rule that time on overnights should be counted toward the 1250. However, how many pilots are going to fork over 10k+ in lawyer fees, for maybe winning and then maybe getting the time off...but definetly being 10k+ in the hole.
In any case, all this precedent is about to become a moot point as Congress has just passed a bill to ammend the FMLA to include flight crews. Obama is expected to sign it into law.
http://blog.aflcio.org/2009/12/03/family-and-medical-leave-extended-to-flight-crews/
As far as "the flu" being counted as a serious condition...The thing is...you don't call HR and say you have "the flu"...you are not a doctor, nor are they. Only a doctor can diagnose a condition. You get the "Certification of Health Care Provider Form", from HR and then you have your doctor fill out the form. The doctor certifies that you have a "serious health condition" and then you turn that into HR...it is none of their business what is wrong with you. If they want a second opinion they can pay to have one done.
As far as the flu specifically, if the doctor says you can't work for 3 days and then prescribes you medication then you meet the definition.
The DOL model form with the definitions written on the back is available here...
http://www.dol.gov/esa/regs/compliance/whd/fmla/wh380.pdf
The main point to remember is YOUR DOCTOR not HR determines if you have a serious health condition.
As far as doctors, I have chosen my current doctor based on his "policies" (--he has none) regarding FMLA certifications and the issuance of generic doctors notes. If your doctor is having too good of a time being the arbitrator of FMLA benefits at your expense...I suggest you do what I did and just get another doctor. Also note that FMLA does not require you get the certification from the FIRST doctor you visited...if the first one won't certify...go get a second opinion.