Why would a company care about its stock price, if it's already
received its money from the IPO?  It seems that stockholders on a
board would want their share's price to keep going up, but aside from
a CEO's being worried about being fired by the board (or the value of
his own shares), couldn't a company just do what's in its employees'
interest and long-term product dev't interest instead of short-term
stock price increase?  It seems that b/c a company really doesn't have
much interest in stock price, that the correlation between stock price
and company performance is tenuous and just based on a sort of social
cooperation among stock traders, and not much else.

Sadly, companies do have to care about their stock prices.

If the stock price is perceived to be low then there is a risk that
predators will pick it up.

Conversely, if its stock price is perceived to be high then it may
wish to become a predator itself.

There is also a lot of 'institutional pressure' for companies to do
the 'right thing' for their stockholders.

More money is made these days by 'the professionals' from acquisitions
and mergers than from anything else.

It's not healthy but that's the way it is.

Ideally, a company has received the mone from the stakeholders and all
it has to do now is perform and distribute the profits to the
investors.
But there is speculative pressure on the company and the bosses keep
wanting to seem like the company is doing great under them. This could
explain a few things.

One way of looking at this apparent paradox is to remember that the
shareholders are the owners, who employ managers to run the company.
So the owners of the company are interested in the value of their
investment.
Management should care about stock price--as well as more long term
factors such as R&D investment--because they can be removed by the
owners.
Also, companies need to raise capital in order to invest in projects. 
There are two ways of doing this:  equity and debt.   Stock prices, as
well as bond ratings, affect the ability of the company to raise
capital.

a company has only a legal duty to care for its stockholders.
Not for its customers, its employees, the general public or even the
american government. that duty is to raise profits and pay a dividend
to its owners.

Sorry, Kemlo, But I must disagree ...

Companies DO have legal obligations vis-a-vis their Employees,
Customers & the Government.

Think about it,

Because the board and the CEO generally have significant stock
holdings in the company, they might be interested in steady or rising
stock prices.  Also, the company itself might hold a number of its
shares... a high stock price might allow it to sell off additional
shares to invest in spending priorities.

Another thing to consider is that in some instances, the management
team is less interested in the actual stock price than it is in
managing the expectations that inform stock prices.

For example, consider that the management team routinely makes
projections about future performance.  This leads to analysts setting
quarterly and annual performance targets, earnings per share (EPS)
projections, etc.  The stock price can react to these projections
positively or negatively, and in this way serve as a barometer of the
market's perceptions of these projections.

But, as others noted... 
rising stock price = happy shareholders = keeping highly paid job.

I should also add, this short-term approach is a problem for some
publicly traded companies.  While it may be good for short-term
shareholder value, it can be damaging to the growth of the company and
sharefholder value in the long term.  As such, it can contaminate
market behavior until that market becomes arguably anti-capitalist.

There are an increasing number of incentives to remaining a private
company that does not have publicly traded shares.