What are the effects of the increase/decrease in stock options after an IPO?
Exactly.  The main issue is if they want to raise capital a second time. The secondary market does not affect the company itself, but it affects its investors. And as investors are important, they need to be treated well.  Another problem related to this is that several employees in the company have stock options, so it really affects them.

What companies can do also is buy their own stocks if the price dropped considerably. This is not something that happens right after the IPO because it would make no sense to return the money to the markets right after you received it.

So, to sum up, the direct effects for the company are on the primary market (when the company makes the IPO), the secondary market is between public investors that have little to do with the company.