Krugman is wrong. In many circumstances, countries are competing with each other. For instance, imagine Kuwait produces oil, and Japan produces automobiles. Kuwait trades oil to country America for automobiles. Now imagine Japan enters the market and makes automobiles that are much better and cheaper than America's. Kuwait will now start buying automobiles from Kuwait instead of America. America will find the price of oil skyrocketing and it's standard of living collapsing, unless it can produce automobiles that match the quality and price of Japan's automobiles.
No, because in my example America has no source of foreign exchange it cannot buy autos from Japan, it has nothing to offer Japan in return (it's a simplified example, but it shows an underlying dynamic that can and does exist).
EDIT: to clarify, there are also examples where increased foreign industry will benefit America. For instance, if Kuwait doubles its productivity, oil imports to the U.S. will be much cheaper. But my point still stands that this is not also the case. Increased productivity from other countries can be positive or negative to America, depending on the specific situation.
Right. To be a participant in a global marketplace means having specialties you can trade on. This is why a common strategy of the emerging economies of the world, past and present, is to heavily protect and subsidize infant industries so that they aren't crushed by foreign competitors. As they establish themselves, the barriers can be gradually lowered until the industry is competitive on the open market.
Free-trade policies tend to be most beneficial to the market incumbents; but I would note that even when a foreign company simply comes in and exploits cheap labor, the workers will get better wages, and even if that doesn't directly benefit their own lives, it can give their children a better lifestyle and education.