As Linkara famously pointed out, Superman unilaterally disarming all the world's nuclear weapons may be a good idea from an idealistic standpoint but it would have terrible geo-political consequences. For example, without the fear of mutually assured destruction, there's nothing stopping the US and USSR from turning the Cold War into World War III.
And the fact that Superman appears to be "getting rid of nuclear weapons" by catching missiles and throwing them into the Sun. Just what were the superpowers doing to launch that many ICBMs in the first place, and how is this an effective means of disposing of all the weapons if these are just test-launches? Are they actually, presumably covertly, in the process of engaging in all-out nuclear war? Or is it just the means by which they are helping him dispose of them?
Linkara was wrong- or at least, greatly exaggerating. For one, the USSR never had the fear of Mutually Assured Destruction- it was an American policy, not a Soviet one; many Soviets even believed they could survive an all-out nuclear exchange. And even though it was an American policy, even the Americans wrote up various plans and tried to find ways that they could "win" a nuclear war, such as First and Second Strike plans (ie. plans to take out the Soviet leadership and / or their nuclear missiles). President's Carter and Reagan were both very sceptical of Mutually Assured Destruction as a policy as well and looked for alternatives (such as the "Star Wars" defence system) and many US military commanders have reported that they never felt that MAD was their only military option. There were several non-nuclear reasons for World War III breaking out, such as the fact that been that would be highly destructive- and more importantly, long and costly-, not to mention that there were two freakin' oceans between each of them, one on each side, which along with the size of both countries meant that any sort of invasion or conflict would be a logistical nightmare. This without getting into how close nuclear war came to actually breaking out on several recorded occasions (which would have nullified the MAD logic), or many other factors.
And as the first episodes of Justice League point out, nuclear warheads are not just weapons but also shields and people can just as easily take advantage of the lack of said shields.
If Superman is willing to disarm the world's nukes, why would he have a problem wrecking all conventional weapons to prevent a non-nuclear war?
For one, it would be a lot harder to do, since there are a lot more of them and they are much easier to make. Plus, a nuclear weapon is far, far more destructive than any conventional one.
As mentioned above: how it catching missiles mid-flight and throwing them into the sun going to get rid of all the world's weapons? Are they launching them just for Superman to catch?
That seems to be the idea, yes. Remember, both the US and the USSR seem pretty happy with Superman's plan, bar a few rogue generals.
If the Daily Planet really was a loss-making concern such that Warfield saw fit to turn it from a 'quality' newspaper to a down-market tabloid, how is it in the banks' interest to invest in order to return it to the way it was before? Good PR?