Brand image is a complicated concept but it consists of a lot of different pieces over a long period of time that build a public perception. Often, good or bad brand images are not due to any one single thing, but the totality of things that the public knows about a brand.

While a single bad apple might not sink a brand, if it starts to become a pattern, it can ruin brand image over time.

Most people: no. Some people: yes.

If Tiger Woods was the only way Nike could think of advertising then they likely would've considered him worth keeping, but when they have so many alternative good options for sponsorships it tips the balance in favour of not sponsoring somebody that even 0.1% of your customers might think badly of you for.

I'm not an expert, but I'd suspect that people were more likely to buy Nike because they cut Woods loose. Lots of press there.

A sport apparel company abandoning an athlete is the type of press that sells shoes?

I'm just speculating from the "no such thing as bad press" dept.