The ethical position of the Torah with regards to cloning, as well as its Halachic ramifications, is of great discussion among contemporary scholars of Jewish law. I will limit this response to the actual question you asked — cloned animals vis-à-vis the prohibition of eating milk and meat.
It is important to note, that at this point in the scientific revolution of cloning, the cloned creature has a gestational mother. Even if the animal bears none of the DNA of the mother, it is carried by a mother and born from a womb.
According to Torah law, if a kosher animal gave birth to an animal which did not have the appropriate kosher symbols, e.g. a cow which gave birth to an animal that does not chew its cud or have split hooves, such progeny is considered Kosher and may be eaten.
Based on the above ruling, the majority of Halachic authorities have ruled that a clone's identity is determined by the mother in whose womb it developed.
Accordingly, an animal born from a cow has the Halachic status of a cow and its meat would be forbidden to be eaten together with milk.
The above information is for research purposes only. As in all issues that are subject to Halachic debate, before acting on such information, one must consult with the rabbi of the community.
I hope that this information is of help to you.
All the best,
Rabbi Baruch S. Davidson
Re: Kosher cheeseburgers Even when science develops a meat that is not created within an animal at all, a cheeseburger made with this meat would be similar to the discussion in Jewish law about eating meat with almond milk. Rabbi Moshe Isserlis ruled that if one eats meat in almond milk, one should place almonds in the area so that no one should think that he is transgressing the prohibition of cooking and eating meat and milk together. This is in keeping with the precept of Mar'it Ayin.
Despite this, many authorities write that since nowadays the use of non-dairy milk or cheeses is so prevalent, people will not suspect that they are actually dairy, and therefore may be used with meat. The same might apply once fake meat earns that degree of popularity, we will have to see what the Halachic authorities rule on this matter.