Is it possible to use Wordpress to build sites, but not host on Wordpress?
Absolutely. Most hosting companies (GoDaddy, Bluehost, HostGator, etc) all provide Wordpress hosting. It's a pretty simple process to get up and running.
Yes
WordPress.com is free website hosting SAAS as in angelino97.wordress.com but you can also plug a domain name in as angelino97.com while being hosted on wordpress.com servers
WordPress.org is the open source software for those that want to get and control their own web hosting
WordPress.com is basically built with the wordpress.org software, albeit a customized version.