Scrape and monitor a company's competitor's job listings for them. Some of that data might be difficult to get given the nature of job sites and craigslist and such, but could be interesting to accumulate all of that (including from the company's own site) so you can get an idea when they are hiring.
Maybe.
If they try to hire for more positions than in the past they're probably growing, conversely they might be stagnating if it's the other way round.
If they hire people with specific skills it might also tell you what they're up to right now like going public or working on a new, supposedly secret project. Take Apple for instance. A notoriously secretive company, previous new projects like the iPhone, the Apple Watch and most notably a self-driving car have first been revealed by their own job postings.