Firstly I am not a DNS experienced person but generally;
A> You need to ensure that the /etc/resolv.conf file in each end-point machine has the correct domainname, domainsearchnames within. Usually your DHCP service would set these up, as I am sure our organisation has done so already.
B> If I understand your point 3. you don't want any other domain to be resolved? The main point of DNS is to resolve domains somewhere along the path, so unless you want to hide a owned domain within your organisation (I couldn't see why) why would you want to you would do it this way? Simply don't add it to your DNS service.
C> Your point 4., should be not a problem but you need to read up on HOW-TO. I strongly suggest you do a google search on Linux DNS HOWTO and go from there.
Cheers.

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Grahame
AMD Phenom(QuadCore), 8GB, 3ware RAID6 1.8TB, HD3850(512MB) ..etc.