Can two companies own stock in each other?

Absolutely. In fact, all stock purchases of more than 5% of a company's stock must be reported to the SEC, so assuming A and B are publicly traded companies in the US, the purchase would likely be a matter of public record.

There are probably special cases where this could cause problems, however; any case where A's purchase of B's stock (or vice versa) runs afoul of regulation would be one such case. For example, if company A wants to own a controlling interest in company B and appoint members of its board of directors and both companies were in the same heavily-concentrated market, regulators may frown on the potential for decreased competition. Such regulations may apply to any purchase of a controlling interest in a company, though.

Yes, this can and does certainly happen.

When two companies each own stock in each other, it's called a cross holding.

I learned about cross holdings in reference to Japanese companies (see Wikipedia - Keiretsu) but the phenomenon is certainly not exclusive to that jurisdiction. Here are a few additional references:

• Investopedia - Cross Holding Definition
• The Economist - Business in Japan: Criss-crossed capitalism
• How does Cross-holding affect corporate governance and financial reporting? [PDF]

I was looking at NAT and NAO, NAT owns 20% of NAO. They trade opposite each other on the price of oil, low is good for NAT, bad for NAO. In bad times the other company's stock would probably rise, so they could trim excess shares to keep a stable monetary holding. This would create cash in bad times, in good times they could buy more, creating a floor as well for the other.

Yes, this happens a lot. And in many cases companies don't even know this is happening.

Collateralized Debt Obligations frequently contain pieces of the same financial products, where it is not obvious what the underlying asset is.

It gets complicated to explain, but I can make an analogy to a portfolio of stocks you might create. Your portfolio contains companies and those companies also own some of the other companies in your same portfolio. The value of all the companies in your portfolio are very interrelated even though you thought you made diversified investments, under the idea that they can't all do poorly at the exact same time. Except they can, if the value of the company's shares are solely based on the value of other company's shares, but nobody noticed that none of them have an actual robust operations.

This was a key factor of the financial disaster around 2008, but this problem was solved with the addition of additional disclaimers that all investors agree to, so they know what they are buying