reenskaug rocker wood type  

Can we see a photo of the whole rocker? There are a few very similar designs that were not made in Denmark. I wonder if this is one.
The light is very harsh in the photos, but the grain in the top view of the arm looks a lot like some legume family tree.

If you want a photo of the complete rocker, that means I have to put it together. 🙂 I have a reenskaug rocker in teak and these two are exactly the same in dimensions, taper, production, I mean everything. Let me try to find the seat as that had some nice variation in color and grain.

If you look at Australian Blackwood and Koa, both Acacias (both legumes), you will see similarities in the grain. I don't necessarily think it is either of those. And I don't think the Danes would have used either, but I think you will see what I mean. Also Guanacaste, another legume:
http://www.wood-database.com/lumber-identification/hardwoods/australian-...
http://www.wood-database.com/lumber-identification/hardwoods/koa/
http://www.rarewoodsandveneers.com/images/productimages/rarewood/Enterol...
There are lots and lots of possibilities.
Maybe with more photos....

Another example: Taverneau aka Wild Tamarind (Lysiloma latisiliquum), another legume:
http://ancientpoint.com/imgs/a/e/r/z/x/mid_century_modern_raymor_arthur_...
It is hard to understand how it could be any of these based on Danish wood selection, but it looks like them. And there are a LOT of pea family trees, with wood that look basically like that.

Does the rocker seem notably heavy, or light? Say compared to teak?

Probably a smidge bit lighter than the teak one.

I was in the lumber store today and came across African Padauk and it was exactly the same wood type as the chair. Not seen many other danish pieces using this wood though.

If it looks like Padauk, exactly like Padauk, then it probably really is Padauk. And, while it is not a usual choice for Danish furniture, it is a reasonably common exotic for studio or hobby woodworking. And I believe this would have been true 50-60 years ago in Denmark as well.
And Padaik is in the Fabaceae family, also known as the Legume or pea family.
If you doubt your own ability to identify woods, then dig up some other pea family woods at the store (there really are lots of them) and see if you think they look the same too. If they all the same, then you can doubt your abilities. If you still think it looks just like Padauk, and not the others, then I say congratulations!