If I own example.com/net/org, can I simply tell my customers to type example into their browser address bar to get to my site? (i.e. no http://, no dot anything)
If I own the .com/net/org versions of my site, can I tell customers to type simply the name of my company into the address bar?
domains
What Firefox is doing is using the first result from Google's search results as their destination. If you don't rank Number 1 for your company name when they do their query your customers will go elsewhere which may be a competitor or a website dedicated to mocking or trashing your company. This can happen as a normal fluctuation of Google's results or because someone has simply done a better job of outranking you. Since you don't control Google's results I don't think this is a very good idea.