"How does one PROVE something which has not occurred?"
I said evidence, not proof.
"The things I am referring to: [...]"
I'll just summarize the categories for brevity: federal mismanagement threatening its viability, economic problems, cultural pollution.
"[...] The United States is not near as strong as it was in decades past, and we just keep getting weaker and more divided all the time."
Many of those problems are ligitimate, but I don't think as severe as you think.
Federal government problems are bad, but there have been worse, and other countries have experienced even worse and survived. One thing about a democracy is that even when it's completely random, useful people will get in power at least half the time. The federal government of the United States has an immense amount of power (more than it was ever expected when it began), but if it looks like it's in a very real danger of collapse, there are too many people who depend on it to allow it to disappear.
Much like the bankruptcy laws which were developed in response to megacorporations allow them to be protected under legal overseership to restructure, rather than letting them collapse under debt and losses - because these days, small nations can collapse from the loss of a Chrysler or Boeing or Time-Warner, it is better to bend the rules in those extraordinary circumstances.
Despite it's offensiveness and waning influence, the world does still need the United States - and the States still need the federal government. You can be sure that most of the nation's resources, and some from other countries, would be dedicated to maintaining a federal government, even to the point of replacing those who had allowed it to get to such a point (or the federal system altogether). Maybe the voters, maybe the Governors and Congress, but it would get done.
The U.S economic problems are actually fairly mild. Especially compared to the Reagan and Carter years during the energy price shock (a situation so bad that four centuries of economic theory was at a loss to explain it). Much of what people complain about is cosmetic, and while there are some fundamental problems, they have all been dealt with and solved elsewhere - the U.S only needs to do so when it gets bad enough that practical considerations outweigh ideology (some of which is the cause of those fundamental problems, through short-sighted selfishness).
Your third point, cultural pollution, has been an issue ever since Americans first complained about those pale-faced clowns coming into the land telling everyone they had to change their ways. However this seems to be more of a personal dislike than the identification of a real problem, apart from the social friction caused by non-conformists trying to figure out how to get along with one another. This is more the result of a wide change affecting all industrialized countries.
Futurist Alvin Toffler identified it in the 1980s, naming it the "third wave" - the first wave was agriculture, which took several thousand years, the second was industrialization, which began in the 17th century. Basically, there are immense and rapid changes due to the emergence of a new range of technology which has the effect of freeing more people than ever before from traditional roles and behaviours.
This freedom will not stop, and everyone will have to adjust to the fact that they can no longer make simple assumptions about other people, or expect implicit agreement on various topics. A lot of what you complain about is simply that that adjustment is still in progress. But a lot of your objections seem to stem from these changes going against what you persionally think everyone else should do (which is a fundamentally intolerant position with no future). In that you join a large (but dwindling) number of people fighting to retreat to a "golden age" of the past, much like the people in that age who thought their own past was much superior - regressionism is a trait of many humans who don't like or can't understand the changes around them, back to the discovery of fire.
Part of the objection to cultural pollution (my term) is also bigotry. I won't accuse you of that.
"The United States has not been so divided a country since the Civil War as it is right now."
The race riots occurred between now and then, as did the shooting of the anti-war protestors, the army attack on veteran demonstrations in the 1930s, and communism was a very real possibility of winning in the U.S during the great depression.
You might mean that the United States is more partisan now than at any time in history, and that is true. I don't know what they end result will be, but I strongly suspect that the far right will wither away, but continue to become more defensive and vocal as it does so until it is reduced to a few violent but ineffective militia groups which will be supressed, and finally lead to the repeal of the right to bear arms. I don't know what will replace it, but democratic reform of some sort will be demanded at the point where violence breaks out, and eventually enacted.
In a sense, the U.S has been coasting on momentum since its last great social change, and is behind the rest of the developed world, but will eventually begin moving again.
"[...] Did you see the scenes of Katrina? That was one, albeit powerful hurricane, and it shredded the fragile civility and order of a major American city."
It wasn't so much a breakdown as a revealing - much of that was already there, as in other U.S cities, but hidden by the more well-to-do layer of population living above them. Many solutions to social problems are practiced elsewhere in the world, but are unpopular in the U.S. I suspect there will eventually be pressure to adopt those solutions (see above).
"Do you not think that, if a serious series of calamities befell the entire United States [...] that Americans could not quickly turn on each other [...]"
I just don't see things ever getting that bad. The trends over the past 500,000 years of archeological and recorded history has been increased prosperity, following an exponential curve. Many people don't know what "exponential" means, using it instead to mean "big", but it has a very specific meaning. Computer genious and futurist Raymond Kurzweil points out that all exponential trends which have hit limits have been followed by replacements which continue the same trends, and that is likely to also continue. For example, oil will likely be replaced by something more plentiful and efficient when it runs out (there are many candidates).
There would still be environmental impacts, but that's a different story (many environmentalists expect the natural environment to be gone by 2075 - species extinction is not the first sign of trouble, it's the last, indicating it's already too late for most of nature, and has been for a couple of decades).
That aside, "exponential" means an always accellerating rate of change, eventually exceeding the human capacity to comprehend it. Opinions vary as to what happens then (Kurzweil believes humans will computerize themselves to keep up).
Finally, just a historical nit-pick:
"Well, we gave up control of the Panama Canal."
Only symbolically. The U.S is still effectively in control of Panama, and always has been since U.S Marines captured it from Colombia. A case in point is the capture of Manuel Noriega. The regime change in Iraq required a complete invasion. In Panama, the military was essentially under U.S command, so stepped aside and put up no resistance when informed that the landlord was visiting to evict an undesirable.