ARE THE ECONOMIES OF CHINA AND THE U.S. CO-DEPENDENT?

In a word, yes. For now.

Generally, every national economy in the world is dependent of the health and success of the U.S. economy. The current global economic problems have their roots in the U.S. What happens here doesn't stay here. As the U.S. economy goes, so goes the world. This is true for many reasons, not the least of which is that the U.S. dollar is the world's reserve currency. Every nation has a stake in the dollar's maintaining its value.
It seems like a long time ago, but there was a time when the U.S. was a creditor nation. Other nations owed us money. We were the world's banker. China was a debtor nation. They were poor. Now, the reverse is true. The U.S. is a debtor nation, and China (and Japan) is our banker. We have to borrow lots of money from them because Americans consume far more government services than we are willing to pay for. China has been willing to finance our consumption because their economy, since some capitalistic principles have been embraced, is generating a lot of money which has to be invested somewhere. Like a lot of other nations around the world, China decided that the U.S. offered safety and a good return, so they have invested heavily in U.S. government securities.
China's economy is export-driven. They sell a lot of goods in U.S. markets. In order to keep the prices of their goods low, the Chinese have for years suppressed the value of their currency. The U.S. has mixed feelings about this. On one hand, the artificially low value of the yuan creates an unfair advantage for the Chinese as the U.S. competes with them internationally. On the other hand, America really needs to import cheap goods. So while we sometimes complain about China's currency not being allowed to float toward its true value relative to other currencies, we don't really mind.
So right now, it goes like this: We buy tons of cheap Chinese goods, and they invest their profits in U.S. government securities. But how much longer can this co-dependency go on?

The Chinese government understands that, in the long term, their economy must become less export-driven and more consumer-based. That is to say that more and more of their goods must be sold to their own citizens. Right now, they have too many eggs in the export basket. Being one of the most populous nations on earth, their potential consumer base is massive. But, at present, the citizens of China are too poor to be a better market for Chinese goods than citizens in the rest of the world.
At some point in the future, China's expanding middle class will be able to buy the kinds of goods that are currently exported. This will diversify and stabilize their economy. Also, at this time, there will be no reason not to allow the yuan to find its own level in currency markets. When the yuan becomes stronger, the costs of Chinese exports will increase for consumers around the world. America will no longer enjoy the cheap prices we currently have at Target.
How much longer the Chinese will continue to finance our government is an open question. Very recently, Chinese government officials asserted that they have no intention to abandon the U.S. debt market. This statement was made despite China's having recently sold some of their debt holdings in the secondary bond market.
So, for now, the Chinese and American economies are co-dependent. China's long-term plans are to de-couple from the U.S. This may take a long time, but the U.S. will have to adjust to China's long-term economic policies.