Domain Theme allows you to specify more then one domain name with your WordPress installation and assign individual themes to each one.
Yes. There are options to enter a Blog Title and Tagline that will override what is specified in 'General Settings' for a given domain name.
Yes. Use the Add Domain form to specify a single domain name and select the theme to use from theme drop down menu.
No. Widgets are defined at the site level and will appear on all supported themes. However, almost anything is possible from within theme templates and you could just alter a theme template to omit or allow certain widgets manually.
You will need to modify your DNS settings (and possibly server settings). This may require you to contact your ISP to ensure that your site allows multiple domain names per server.