Can I post an online company directory freely?

The companies have the right to control their trademarks, which often include their names. Also, a lot of companies like to keep their addresses and contact information closely held so that they are not bothered by people who waste their time or endanger them. If you do make such a directory, please be decent and have your own contact information posted so that if people want their listing deleted, they can easily contact you. Then make deletions easily and quickly.

Yes, essentially you're creating the electronic version of a telephone directory.

You can collect and post the information. Much as you would the yellow pages. There is no copyright to listing information.