Yea you are right. I was just trying to make a useful comment since your questions were very broad. Let me try again:
1) Where to find such investors? Are there actual people looking to invest their money in web projects?
There are angel investors, VC's, and then random things like family investors, etc. There are networking events for this kind of thing, and people are pitching their dotcom ideas to investors all the time. Yes, there are large numbers of people/organizations who investing in online businesses (not a website, not a project, but a business)
2) What does an investor expect in return? Imagine you have a web project, but you're not on any stock market with your project (or not yet...). What are the advantages for the investor? How does he earn his money back?
It depends on what you have to offer, what you bring to the table. If your idea is fantastic, you have great business skills, and just need some cash to turn the key you can get money without giving much equity away. If not, you may have to part with big chunks of equity to make it worthwhile for an investor.
3) What exactly does a site do with the money they get from an investor? They spend it on advertising and employees and such?
This depends on the business. An ideal investment would be one that already has a great team, a great plan, a great beta/prototype, and a great marketing plan that just needs to be funded.
4) What if the project fails? Is this a risk an investor takes, that he could lose his money?
Sure, they are risking they money by investing in a startup. That's why they will want a lot in return.
5) Do you know any other, small and big sites, that are sponsored by an investor?
Yea, a significant percentage of them are. Businesses like Facebook had early investors, and if you look around on the web for venture capitalists you can sometimes see their portfolio of sites. It's very common.