Yes, you can use your own domain name to access Odoo (ideal for the Website). You need to contact online@openerp.com and ask them. See also the section: "Can I use my own domain name? And also for emails?" at https://www.odoo.com/page/pricing

You will need to enable port forwarding to have your site  redirect port 8069 to the IP (222.444.555.666:8069 forwards to 222.444.555.666 using apache server or nginx on Ubuntu). You need to make sure that your database has the domain set at your mydomain.com instead of localhost for web.base.url. You can find this in Settings>Technical>System Parameters.

Hello,

I have exactly the same issue, me using Enterprise 10.

I do not find any info about on Odoo official docs.

How to add custom domain to my local enterprise 10 installation ?

I currently have website displayed default way:

http://222.444.555.666:8069/

I would like to get website displayed on http://www.mydomain.com/

Thank you