JJ-
The doctors who told you to take iron was making a conceptual error. Yes, people who live at high altitude do get thicker blood, but that takes a long time. Certainly longer than a couple of weeks, IIRC.
Now, speaking as a specialist who has been out of medical school for 7 years and has forgotten a lot of this useless stuff…
Acetazolamide (Diamox) has been shown to speed altitude acclimatization in that you will achieve a higher blood oxygen level a little sooner if you take it. This is not primarily due to it's diuretic effect though, as Mark was alluding to. (Though, of course, being a diuretic conceivably helps with HAPE or HACE. But there are much better things to do for those, like GO DOWN THE MOUNTAIN.)
Acetazolamide somehow speeds the adjustment of the threshold for respiratory stimulation due to blood PaCO2.
For you laymen, here's a thumbnail sketch:
One of your body's stronger stimuli to breath is a high blood CO2. The CO2 level in your blood is normally essentially independent of the O2 level in your blood. But here's the thing- the faster you breath the more CO2 you will off-gas and your blood CO2 level will go down! (Well, technically, it isn't how fast you breath- it's your minute ventilation, which is breathing rate x breath volume. Sort of.) When you go to altitude the CO2 in your blood is essentially stable so you still breath at a low rate just like you did at sea level, but since the air is thinner you are getting less oxygen. Thus your blood oxygen goes down, and you feel weak and tired, and this somehow leads to AMS, HAPE, and HACE. You body will eventually realize this and adjust the threshold for respiratory stimulation to a lower CO2 level, so that you will breath faster to get more oxygen even when your blood CO2 is low. Acetazolamide somehow speeds this adjustment. Typically the dose is 125mg at the start of the elevation gain, and 125mg after 12 hours, IIRC, but I'm not sure. EDITED– UpToDate says 250mg prophylactically every 12 hours starting 2 or 3 days before the trip for rapid ascents to over 7000-8000 feet and stopping once you've been at goal altitude for a day. It contradicts itself on effectiveness after symptoms have already appeared.
So, I THINK acetazolamide makes you feel better and perform a little better due to higher blood O2, and i think it reduces the risk of AMS, but I don't think it has been proven to reduce HAPE or HACE. I'll have to check when I get home. I do recall that it has been proven to greatly improve sleep quality at altitude, as Mark alluded to. (Bad sleep is caused by Periodic Breathing of Altitude which, believe it or not, is a type of Cheyne-Stokes respiration.)
This is all similar to what we think causes shallow-water blackout, by the way: Typically, someone breaths fast and deep for a little while in an attempt to oxygenate their blood so that they can stay underwater longer. In reality, they have only driven their blood CO2 level down. (For all practical purposes you can't get blood O2 above 100% hemoglobin saturation. There is a small amount of O2 dissolved directly in blood plasma, but it is a trivial amount.) Due to the low blood CO2 they don't feel an urge to breath! Eventually, if the O2 runs out before the CO2 climbs high enough to drive a respiratory urge, they pass out and drown. This is why experts advise not taking more than three deep breaths before free-diving.
By the way, you loose you acclimatization VERY fast. There have been people who live high in the Rockies who have gone down to town for only 2-3 days, then gotten AMS upon returning home. Well documented.
The nice thing is that you shouldn't have much trouble talking your doctor into prescribing you some acetazolamide if you tell him you're going mountain climbing (or whatever). It's pretty benign, but it counts as a sulfa drug, so if you have an allergy to sulfa you can't use it, and it plays hell with blood sugars in diabetics. You might have to give doc some time to read up on it, though, before he'll write you the script. On the other hand talking him into giving you some dexamethasone will be difficult, to say the least, even though the studies show that it is a better prophylactic than acetazolamide. You'd better provide PROOF of your upcoming Everest attempt…
And, dammit, it does ruin the taste of ANY carbonated beverage. I think it inhibits carbonic anhydrase, or something, so things tend to taste more acidic and sour.
Reference: Auerbach's "Wilderness Medicine," currently in it's fifth edition, though mine is older. This is a $160 textbook that isn't of much use to laypeople, so though it does sound interesting don't rush out and buy it. It's very technical.
Also, yes, Advil and other NSAIDS (like Tylenol, Motrin, etc.) act as "blood thinners", but that is a lay misnomer. These drugs slightly inhibit clotting, and this is what is meant by "blood thinner." They don't have an appreciable effect on the number of blood cells per unit volume of blood (i.e. hematocrit or hemoglobin levels). Confusing terms.