1. Why create a child domain? corp.mycompany.com .. don't get that design and wish to GOD that MS wouldn't teach it like that.
2. Who is the SOA? If you've given SOA to the hosting company, you'll need to change that with the registar service you used to reg the domain name.
Suggestions...
mycompany.AD or Corp or fartknocker ... the reason I say this is that AD does not require RFC compliant top level domain extensions.
When I design em' I keep the name and add .ad or .corp. The primary reason is to avoid confusion with DNS, e-mail etc. So I'll have an outside domain name of mycompany.com and an inside of mycompany.ad
Personally I'm all about keeping it as simple as possible, especially in my naming conventions.. the .AD reminds me that it is the Active Directory zone and that way I don't do something REALLY stupid to it.
You'll need two DNS zones for the example I'm goiing to give here; if you're going to host the "real" domain on the same server as the internal one you'll have to have a second NIC (OK well not really, but it is a REAL good idea to have the second NIC rather than just assigning the IP to the primary NIC in addition to it's internal IP) and punch a hole in your FW to foward DNS calls to that server. A better practice is to have 2 DNS servers, inside and outside. Neither has to be dedicated or real ballzy .. but you'll find DDNS functions rather well when it's run from a DC rather than a member server.
The outside server can be of what ever flavor you wish, the internal server should have a fowarder pointing to the external server so that all resolutions that are not part of the internal zone go out first to a server you manage, then up the root hints line.
damn I hate DNS