The text for cultists with Ithaqua states that if an investigator loses health on the strength test, they gain a hypothermia condition. If you get one success, you fail the test since the cultist has 2 damage. If you prevent the health loss from another investigators spell, do you prevent gaining the condition, even though you failed the test?

I don't know what the "correct" ruling is on this, but when in doubt play exactly as the cards tell you.

By this I mean, yes... if you do not lose health then you would not take hypothermia.

Preventing the health loss does prevent the hypothermia. It does not matter if you fail the test.

Remember, your "test result" is just how many successes you roll. You compare your test result to the monster's score and if your test result is lower, you lose sanity or health equal to the diffetence. (See https://community.fantasyflightgames.com/index.php?/topic/133222-eldritch-horror-combat/ for details).

The only time you need to care if you pass or fail a test in combat is if there is an effect that says "If you pass the (attribute) test, then..." or "If you fail the (attribute) test, ..."