Is it possible to make to a fully functional custom player race using the Creation Kit?

I have created a custom player race that plays fine. But I am sure that it's not 100% functional because of NPC reactions lacking race-based reactions to my character.

When customizing this race I used a duplication of the NordRace found under Character>Race section. Should I be using something else, or linking more than one object file?

Yes it is possible to make a fully functional race. However making a custom race that doesn't break other custom races is a little more difficult. To that end you could look at the RaceCompatibility mod. I've made a Dawnguard compatible version here (http://skyrim.nexusmods.com/mods/24168/) and a few fixes for the vanilla version as well. This mod tries to prevent issues with multiple custom races modifying the same resources, and if you follow the instructions your race will function properly with vampirism, certain Dark Brotherhood quests and hair and face part mods.