augury, foreboding, harbinger, indication, portent, premonition, presage, prophecy, straw, warning, writing on the wall*, auspice, bodement, boding, foretoken, prognostic, prognostication
omen
indication, sign, portent, forerunner
Conjecture, omen, prophecy, augury...
Sure, a synonym for "auspices" could be "patronage" or "support."
alerts, warns, admonition, counsels, forewarning, hints, omen, sign, tip-off
Thesauruses and dictionaries do not seem to mention what the antonym of an omen is, but they do mention that and omen is a "sign of something to come." So, if an omen means to predict something that has not happened yet, then the opposite of an omen would either be to predict what has happened long ago or something that has already happened.
A bellwether is that which indicates or predicts a change or a trend. Therefore, synonyms include an indicator, a predictor, a sign or (if trending toward something negative) an omen.
There have been at least five Omen films. The Omen(1976), Damien: Omen II (1978), Omen III: The Final Conflict (1981), Omen IV: The Awakening (1991) (TV) and The Omen (2006), which is a remake of the first movie.
Omen
It depends on context. "The meteors were an omen of pending disaster" or "the patrol's appearance was an omen of more to come" are both acceptable.
Ominous: [Latin, ominosus < Omen] Of or being an omen, esp. an evil one. So, yes, ominous is the adjective form of omen.
There is no evidence that skunks are an omen of anything.