It will reduce them some, but I don't think it will reduce them a whole lot.
A packet capture would show how many you send and receive. However, over a VPN connection it should not be receiving a whole lot of broadcasts.
If you computers default domain name is the same as AD's domain name you can still just use the host name, but using FQDN would be a bit faster. Using just the host name your computer will send two name look ups, one with just the host name, then one with your computers domain name appended to the end of the host name. So you have to wait for the first one to fail before the second one is sent.