As the question mentions, good situational awareness is an excellent safety tool. ATC is responsible for runway separation at towered airports, but pilots can play an important part of backup in the case of a human mistake.
The United pilots may or may not have been able to see the Atlas 747 on TCAS, as the terminal building that obstructed their view could ...
In order to increase the number of operations that can be handled at a runway, you want them configured so as to allow "simultaneous independent approaches," which means that the 2 (or more) runways can be operational without the need to coordinate between them.
The biggest factor is the distance between the two runways. The greater the distance between ...
Specifically, do airline pilots have any additional means for
determining the position of other aircraft on intersecting runways
aside from relying on ATC, listening to the Tower frequency, and
looking out the window?
Basically the answer is no. The only thing that could have been of help is TCAS, however:
Aircraft on the ground are not visible on ...
That division is applicable only when the parallels (4's or 22's) are in use, which is the preferential system (see title of chart).
As for rwy 12/30, from the eAIP (PDF; link may die; homepage):
Between 6 AM and 11 PM local:
RWY 12 and RWY 30 may be used when one or both of the preferential runways cannot be used (...)
Between 11 PM and 6 AM local:
Ideally runways do not intersect each other, and are all 4300 feet apart (laterally) to allow for simultaneous parallel instrument approaches. Denver is one such airport that meets all of this.