Some monsters say if you fail the horror check, then there are consequences. Do these consequences occur if you get zero will test successes or if you take even 1 point of sanity damage? Ie if a monster has 3 horror, do you have to roll 3 successes to pass the will test?
You only need one success to pass a skill test. There can still be consequences (Sanity or Health loss for instance).
If the text says that if you fail the horror check you will suffer blablabla, then you only need your test result to be 1 or higher to avoid the blablabla consequence.
Here are extracts from the Reference Guide to cover those answers (emphasis mine):
On page 11 (Tests):
The third step says:
Determine Result: Each “5” or “6” rolled is one success. The total number of successes rolled is the test result. If he rolls at least one success, he passes the test. If he does not roll any successes, he fails the test
The fourth step says:
Resolve Pass or Fail Effect: The consequence of passing or failing the test is determined by the effect that triggered the test.
- Some effects specify only a pass effect or a fail effect. If an investigator passes a test that does not have a pass effect, there is no effect. Likewise, if an investigator fails a test that does not have a fail effect, he suffers no ill effects.
- On encounter cards, tests are declared using their skill icon embedded in the text in parentheses. The test declaration may also indicate a test modifier.
- Effects that are dependent on a test are always preceded by “if you pass,” or “if you fail.” Any effect in that sentence occurs only if you pass the test or fail the test, respectively. Any effect after the period or part of a new paragraph, is independent from previous effects