Does transplanted heart have full sympaticus without medication?
The parasympathetic supply to the heart is by nerves and that is truncated during surgery as pointed out.

The sympathetic supply to the heart is by two means: 


Through sympathetic nerves from the sympathetic trunk
From the adrenal medulla which secretes both adrenaline and nor-adrenaline directly into the blood stream


The heart contains the beta receptors for the sympathetic system, especially the ventricles and nodes (SA) and conducting fibers. These receptors respond to the presence of adrenaline and nor-adrenaline in blood. Thus the sympathetic response is preserved while the parasympathetic responce is lost. 

There is a minimal level of nor-epinephrine in blood at all times which thus influences the heart. 

Receptors in Heart:  

The heart contains adrenergic (sympathetic) and cholinergic (Parasympathetic) receptors. When the heart is transplanted, the donor heart also contains the same receptors. The only thing missing are the nerves.

The sympathetic system does not suffer as much as the parasympathetic system, because of the adrenal medulla which secretes the nor-epinephrine (same as nor-adrenaline) into the blood, which upon reaching the heart can stimulate the beta receptors on the heart.

There is no organ which releases acetyl-choline into blood, (fortunately, as such a release would be disastrous leading to shock and rigor, possibly death). So the parasympathetic receptors in the heart do not receive any stimuli.

Edit 1:
In response to your edit, yes there are polymorphisms in all receptors. But the structure of adrenaline and nor-adrenaline are pretty much constant. This means that the donor heart will be at a similar level of sympaticus as it was in the donor when (s)he was alive. Although this will not be the same as the recipients pre-surgery sympaticus state of the now removed heart.