Networks

You can use mydomain.com. Companies do this all the time. Your site should be protected from the internet and in a dmz. This is assuming you are not using the website's server for authentication-a/d for your internal domain. That would not be advisable.
You won't have any problems with name resolution, because that is done externally through the company hosting your records on the outside, so to speak. Internally, your server, if on the same domain, would have its own internal host record, independent of the external dns.

You can create a private network by using a dedicated server provided by XnYnZ.com .The full server will be dedicated to you so that you can create your own private network.

Let's say your public domain (website) is: example.com

You should avoid using just that for any internal active directory network (it will cause you problems later on with DNS names, and possibly leak internal FQDN/records to the Internet which is a security risk).

When setting up active directory you can still use your public domain, but the AD should be a separate zone, this is often referred to as NetBios domain name, or pre-Windows 2000 domain name, for example...

internal.example.com or private.example.com

You can also use any of the following non-public and un-registerable root domains like...

.test
.example
.invalid
.localhost

...or with a NetBios domain...

newco.test
newco.example
newco.invalid
newco.localhost

In any case, the second "zone" (really a sub-domain) is what will show up in the "logon to" drop-down box at CTRL+ALT+DEL, for example...

FQDN: internal.example.com Logon domain: INTERNAL
Full computer names: pc1.internal.example.com, pc2.internal.example.com, server5.internal.example.com, etc.

FQDN: newco.example Logon domain: NEWCO
Full computer names: pc1.newco.example, pc2.newco.example, server5.newco.example