Early in the movie, the General says that the launch codes wont work unless they are at defcon 1. When they decide not to Nuke 'em at the end, why didn't he just change the defcon to 5 so that the computer wouldn't be able to launch the missles? Of course, doing so would ruin the Science Is Bad Aesop.
They simply couldn't - WOPR was partially responsible for establishing current Defcon level, so the humans really fell off the loop of control.
The back door password was "Joshua" because the guy who made the computer had a son named Joshua. Because of this, David often refers to the computer as Joshua, even after he discovers that its real name is WOPR. Later in the story, Professor Falken refers to the computer as Joshua. Why would he refer to his computer by his dead son's name?
Turning into Fridge Brilliance: It's Rule of Symbolism. It's almost like Frankenstein. The "father" abandons his "son", who's turned to darker purposes, only here, once his "father" acknowledges him for the first time in many years, the son returns from teetering on the edge of destruction.
Even though acoustic modems were outdated by 1983, modems in general still weren't cheap. An acoustic, likely scavenged from a local university, might have been all David could manage to get ahold of for his obviously-cobbled-together rig.
The two soldiers at the beginning of the movie apparently didn't know they were in a simulation. When one of them wouldn't turn his key, the other threatened him at gunpoint (apparently according to protocol) to follow orders. The follow-up discussion implies the first soldier continued to refuse. We're not told if the simulated nature of their situation was revealed before the second soldier could follow through on his threat.
Both soldiers were shown in a later scene showing the operators' seats (and other relevant human equipment) being removed in favor of the new WOPR automated parts.
There were several non-nuclear war scenarios listed as "games" that David could have selected, including chemical and biological weapon attacks. Presumably, WOPR has a say in coordinating those for the military also, else they probably wouldn't have been listed. If David had decided one of these options looked more interesting than simply nuking the hell out of everything, could WOPR have set off a Real Life military conflict, by faking reports of the use of such weapons? It's a lot harder to verify whether gas or germ attacks have been used at ground-level than whether or not a city has vanished in a mushroom cloud, so if WOPR- had been playing such a "game", American troops might've actually been dispatched to investigate such a fake report, and the place where the "attack" allegedly happened could have interpreted their arrival as blatant U.S. aggression. Granted, it wouldn't have risked the whole planet, but people might have really died in such circumstances ... and WOPR possibly could find a winning scenario for such a "game", so might never have stopped "playing" it.
How bad would things have gotten had David decided to play as the Americans instead of the Soviets?