I think we are confusing some things.
First of al we have ghul'dhan the orc warlock. after dieing and being ressurected by Kil'jeaden, was it? he was known as the Lich king. KJ wasn't happy with him and locked him up in ice. Only the newley reborned lich king put his sword with magic away. Where after time arthas will find it.
Than second of all we have arthas. Knight of the silver hand and prince of lordearon. he is pure good but because of this he hates the undead to much. so he is on a rampage to kill the scourge. but this makes him blind. And gives him control over frostmourne. here his stupidity transforms into madnass. Arthas is know hearing foices. But i don't think the lich king was directly speaking to him. But just sending minds. So arthas didn't hear words. But saw events and opinions of the lich king. But battle is hard so arthas was tired. And then it is hard to know what your own opinions are and that of your mind controlling freak and where you don't kknow of. It's like you really hate someone you wan't to hurt him(not your own opinion, just a situation where you are in, maybe tomorrow you will be best friend). But you are still hurting him. In a relaxed situation you will greave and think about what you have done. But arthas was in war so you can't sit still. you can't go against the flow of maddnes.
But after time this madnas increases and utility he heard voices. (thats when arthas lived alone. when you finished human campaign and then start undead campaign). he than meets a dreadlord (can't get one the name of him) who sais he has a salvation for him. because arthas has only felt anger, yealosy, greave, hatred, lust, hunger, almost death, deseases, revenge and pain) i think he will be pretty tired. so he starts doing everything as long as that voice shuts up. I personally think that if he would have had good company or tried harder to fight te lich king mental. He would have destroyed frostmourne and never become lich king and we wouldn't have this discusion. and i would have spend this evening on something else.
so arthas starts traveling and listening to the voice and becomes more evil every day.
Than he has his little showdown with illidan stormrage and wins. (self speculation on small detail). At the moment illidan lies at arthas feet as the mighty demon/night elf he is arthas had one little moment where he realises what he has done and how he can stop this. Bu it was just a short moment not even a second. and the lich king's iron will strangles around arthas mind. Bestowing him with fallacies on completing the circle and human cowards and perfect death. Arthas not knowing what to believe just follows his master again because it worked before. if we both look at the spirit or mind of both the nez'ghul and arthas we see that that of nez'ghul's is much bigger and stronger than arthas'. So when they emerge it is like you put two liquids in a cup. you can't possibly know what liquid was from what cup. only than you have one cup with a really small amount of liquid and a lake full putting together. I think the lich king know is so a little bid different from the lich king before the fusion. i think less patience but for the rest the same. And arthas' spirit is no more. But one big difference the lich king now can walk. He has a body. arthas' body. When arthas said he intended to live forever. He really mend that. But after the fusion the lich king had other plans with his methabolism. Because when u live u can die he killed himself. So nobody else could do that. Making him again much stronger because his spirit is held in an undead body. and undeads don't feel the cold of northrend. so not arthas ripped of his own heart. The lich king ripped out arthas' heart. The only salvation arthas has left is when the lich king dies (by random noobs in wow) and that he will be ressurected the holy way so a body comes back to live with his originall spirit. And feeling again the warmth of life through his veins. if his body will be ressurected the necromantic way he will just be a ghoul to serve his master. and arthas' soul will stay in heaven. while for ghul'dan his spirit will return to the way it was before the pact with keal'jeaden. And the roots this demon has set will disapear. and the lich kings armour just really good armor. Wich gives epic boosts. a littlebit better than tier 8

maybe death knight gear tier 9 haha.
Sorry for any bad english spelling
awnser on question. Arthas his body is undead as soon as he rips of his own heart and becoming a true
death knight. Because everybody can learn the spells, but just a vew will be unliffing.