"Morgoth's Ring" or Sauron's Ring?

[Fingolfin II]

So partly for this reason it seems somewhat strange that Sauron would be able to increase his power by making the Ring, without the 'extra' power coming from some other source (such as Morgoth's dispersed power).

Sauron let a lot of his power pass into the making of the Ring, in order to control the very powerful Elven and other rings. The 'extra' power that he would have would be due to his dominance over the Free Peoples, because he had made the One Ring in the first place for this very purpose. However, I certainly do believe he learnt a lot under Morgoth, which led to greater knowledge, and ultimately greater power.

Sauron having the Orcs and Easterlings under his sway is very interesting and seems to support your theory about him gathering sourcing some of his own power from Morgoth's dispersed power- why else would they go to him?

I suppose you can argue against that, that they would go over to his side because of fear of him, or maybe because he bribed them (most likely it's a mixture of both). However I think that Morgoth's will is definately prevalent in this matter, as 'the bonds he laid on Sauron' (somewhere in the Silmarillion/Akallabeth) were very strong, so Sauron turned once more to evil and the 'Dark' power he most likely learnt from Morgoth. So in a way, I believe that Sauron's power is coming indirectly from what he has learnt of Morgoth, rather than from Arda itself, since Melkor is outside of it in the Void.