The host server should have its own domain name. You have to apply to your domain name provider to create the nameservers using that domain name.
All subsequent domains you wish to have use those nameservers should be registered using those nameservers at the domain registrar.
Plesk should have a panel that allows the definition of the nameservers for the entire host, use your registered nameservers there. Each additional site you create should then use the defaults.
Individual sites do not need a nameserver entry, only the hosting site for the whole box needs those entries.
You should restart bind after each modification, but plesk should do that for you.
Essentially you should end up with bind entries for each domain that fit the following pattern :
domain1.com A your.ip.address.here
www.domain1.com CNAME domain1.com
domain1.com MX 10 mail.domain1.com
mail.domain1.com A your.ip.address.here