i think a more pressing argument is the lore perspective; you can make a case for a death knight having enchanting abilities, blacksmithing, even leatherworking (and definetely alchemy). JC is a bit of a stretch, but possible as well.
however, cooking, first aid and fishing just dont sound right. it makes sense that a once-trusted servant of the lich king would learn how to enhance their own fighting ability with items or enchants of great power, but i somehow dont see battlefield triage as something a DK would really care about. why heal your minions when you can just slaughter more people and make new ones? more than that, what DK would have the time to fish, or cook? fishing and cooking dont exactly further the cause of the lich king, unless of course the lich king is trying to take over the aquatic realm and wants to start with sagefish and firefin snapper.
that, or the lich king has a weak spot for sushi. but thats only a rationale for fishing.