As written, contingency triggers as soon as its condition is true, not as soon as the caster is aware that it is true. Neither caster’s senses, nor anyone else’s senses, are involved: it just happens. The spell changes the fundamental laws of the universe such that the next time X happens, spell Y is cast.
So the question isn’t about who knows, it’s about what is fact. Going down the list:
1. tell time?
Contingency certainly seems to be capable of responding to certain times (time of day, time since an event); the tricky part here is establishing the condition clearly. Many settings don’t have rigorously-defined times, so it’s not clear that “6 PM” is going to mean anything in-character. On a globe, time zone issues can also be problems. But something like “noon at Town” seems likely to work.
2. read minds?
The open-ended nature of the spell leaves that conceivable, but I’d go with “no,” largely on the basis that personal thoughts are not “facts of the universe” and detecting another’s thoughts requires a Saving Throw. Contingency does not have a built-in read thoughts.
3. predict the future?
Similar to how I feel about reading minds: predicting the future requires special, powerful, and uncertain magic. The future is not a fact, not yet, and so contingency cannot be triggered by it.
4. penetrate disguises?
If the disguise included some form of hiding from magic (nondetection), then no, but a simple mundane disguise would not affect contingency.
5. discern the morality of actions?
Magic can detect alignment without giving the subject a Saving Throw. Somehow, alignment is an objective observable reality. So things like “a creature of Evil alignment” and an “Evil action” could probably be valid. Whether or not alignment is the same as morality is debatable and beyond the scope of this question.
6. employ senses the subject lacks?
Contingency doesn’t “sense” per se; it just automatically reacts to facts. So, basically, the subject’s lack of a particular sense doesn’t come into play.
7. determine other creatures' intents?
Same deal as thoughts; probably not.
8. hear or see things the subject can't?
Same as 6; it doesn’t “hear” or “see” but can certainly react to facts that the subject cannot hear or see.
9. know if the subject's been robbed even if the subject doesn't?
10. otherwise be triggered by information the subject doesn't or even couldn't have?
The subject doesn’t come into play; it’s more about what “magic” could have the information. Protected information (e.g. stuff magic would require a failed Saving Throw to get) I’ll call unavailable, but even that might be wishful thinking on my part.
In short, contingency is horribly-broken as written. You can try to rein it in by houseruling in some limitations, but I’m not convinced they’ll be enough. The spell is almost-certainly the most powerful that Wizards ever wrote.
As for the timing of simultaneous events, the rules don’t really handle that well. Presumably you could just use whoever’s player declared the action first, but in the case of purely-reactive contingencies, both could be triggered by the same fact and therefore be “simultaneous.” For creatures’ actions, I’d consider houseruling an expansion of Initiative to use opposed Initiative checks in such cases. But contingency is not a creature and once cast, it doesn’t have anything to do with the creature it’s been cast on until it triggers a spell on that character. I think perhaps the best solution is to use opposed Caster Level checks?