I had always assumed that after you adapted to the thin air, the risk
to your body was how thick your blood got with the production of extra
red blood cells to compensate for the lack of oxygen.
an increase of red blood cells will only happen after a long period of time. This is why athletes often train at high altitude. this is a very gradual process though and is not to be confused with what happens to you on a high altitude trek. On a trek you body will not produce any significantly greater number of red blood cells (dependent on time spent at altitude obviously).
Also, more red blood cells does not equal increased blood clotting. Blood clotting is performed by platelets not blood cells. Totally different mechanism.
The thicker blood making frostbite and edema more likely because of
poor blood flow, and heart attacks and strokes more likely because of
increased chances of forming blood clots
See above. More Red blood cells does not make your blood thicker. It increases its ability to carry oxygen. For "thicker", don't think "syrup"; think "marbles moving though a smooth tube". More marbles doesn't make it more likely they will get stuck.
I think you're getting confused with Deep vein thrombosis which is caused by inactivity in a pressurised cabin (i.e. plane). High altitude does not increase your likelihood of getting blood clots.
But my friend said it had less to do with thick blood and more to do
with the lack of air pressure...
He's right. The higher you go the less air pressure there is (oxygen remains static as a percentage of air regardless of height). Less air pressure means less air, which means less oxygen. So every breath contains less oxygen the higher you go. So you blood has less oxygen in it because your body cannot receive enough oxygen from its surroundings.
Frostbite is caused by tissue not receiving enough oxygen, i.e. it's so cold your body diverts blood to your core to keep you alive, potentially sacrificing a finger in the process. If your blood has less oxygen because of the above, you're more prone to frostbite because it takes a lower amount of blood starvation to reduce the amount of qxygen in the tissue to the point where frostbite occurs
HAPE and HACE
HAPE and HACE are slightly different to frostbite. They are still caused by the lower amount of air pressure, but no one's 100% clear. You can have two people climb the same mountain at the same rate and be of the same fitness, one may get HAPE and the other won't, obviously something about the person is involved. It is known that lower air pressure causes HAPE and HACE. It's thought that it's to do with the pressure imbalance in the air. Put overly simplistically: Your lungs are designed to work at x pressure, they have difficulty at x -10 pressure.
Having increased blood cells in your blood may or may not increase you're chance of getting HAPE or HACE. It's been known for Sherpas to get these issues and they have a much greater ability to handle altitude than your average western climber.