Melkor is canonically attracted to Varda, Arien, and Lúthien. I'm not suggesting that this would be other than you said - wanting to use or claim something for his own, not exactly like he would go off courting or something! More that he wants what he can't have.... And he does pour his own native power into things to create armies and such.
I don't want to give any of the Valar children, so I don't want Melkor to have children, either. I do think that the Valar/Maiar relationships are interesting, so I'd like to use that conceit in determining the rivalries and interactions in Angband. There *is* a difference in how and why Gothmog (and the other balrogs) follow Morgoth, and how Sauron does.
The balrogs have almost fanatical loyalty. They have been associated with Melkor since the Music, so they are on board with his view of Creation. They are very 'sir, yes sir!' about any orders from Morgoth. It should not be difficult for the viewers to draw parallels between the Sons of Fëanor following their father, and the balrogs following Melkor/Morgoth.
Mairon/Sauron, on the other hand, was originally a Maia of Aulë, and spent most of the first season of this show trying to reconcile Melkor and the Valar. He was acting as a go-between, and Melkor (perhaps confident that he'd win him over in the end), never made any demands of loyalty from him. By the Season 1 finale, Sauron has made his choice (and he obviously admires Melkor's power).
I think we are going to portray a Sauron who has second thoughts about Morgoth's ability to fulfill the promises/dreams he bought into from Melkor....but that it will take some time for Sauron to get them. Right now, he's very much bought in, and trying to make the most of the situation. I think it is only as time goes on and Morgoth weakens (while Sauron's own corruption becomes more apparent) that we'll see Sauron reach the point where he thinks he could challenge Morgoth for power.
In the end, we are probably going to have a Sauron who thinks himself too weak to achieve his own goals, and has a weird worship/hungry relationship with power. This won't really come into play until the 2nd Age, when he makes his own bid for power and crafts the One Ring.