'mail.mydomain.com' is just a record.
'www.mydomain.com' is just a record.
A subdomain is when you have a set of name severs delegated for that part of your domain.
'mydomain.com' is a sub domain of 'com'
You can have 'east-coast.mydomain.com' be a subdomain of 'mydomain.com' but then you would create NS records for that subdomain in 'mydomain.com' and then create your configuration on the new name servers.
But to answer your question quickly. Yes, you can have as many hosts and subdomains as you want. They can all point to as many different IPs as you want. It's all DNS.
You can either setup DNS yourself or use an outsourced provider (either free or with fee).
So, I have www.mydomain.com all working fine. Hosted on linux server. Now, I use a link to an IP address on my network in my office. Are you saying I can just go into control panel, create www.newlink.mydomain.com and add an IP address and it will be directed there?
But, can I have a subdomain from there pointing to a completely different IP address?
If I understand your original question correctly, yes you can. For instance, if you are using cPanel as your control panel, create a subdomain. Then to redirect it to a completely different IP address, click on the Setup Redirection in the subdomain area and enter the destination URL.
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If you are on a dedicated box then just edit the IP in the zone for that A record. It may take from 15 minutes to 1 hour for the change to take effect depending on how your DNS is configured and if you have already tried looking at x.domain.com before the IP change as it will continue to point to the old for a short while.
If I type in the IP address into my browser window, I cannot see the page as I am inside the network. If I have a subdomain pointing to it, will that somehow work or will it not work because I am still inside the network? i.e. does it work kind of like a proxy?