[Mumriken]

English isn't my first language so excuse my english anyway...

As I think has been made clear from quotations, Sauron did join Morgoth initially in admiration for Morgoth's ambitions and methods, but as Morgoth's motivations waned towards all-consuming destructive hatred perhaps Sauron's loyalty did as well.

Perhaps but remember Morgoth wasn't a nihilist at all times. Once he also wanted to rule over others. Also if by the end of his time in middle earth morgoth was a nihilist and truly wanted to destroy everything. Then there are two questions we must ask ourself:

1. Why did he not do so?
2. Why did not Sauron do this?

Is it possible that if Morgoth wanted to destroy everything, the reason Sauron did not follow in his footsteps was because he was unable do so? Sauron was far from as powerful as Morgoth, that is why he created the rings to control all others. Unlike Morgoth who could actually challenge all others and win! So I think it's a bit unfair that you say that in order for Sauron to have been loyal to Morgoth he must have done these things:

-Tried to free Morgoth (Impossible)
-Tried to destroy everything (Impossible for him)

You make it impossible for him to have been loyal to morgoth by saying that.

What's more, he even shows disdain for Morgoth - as some of our quotes from Professor Tolkien established, it seems that Sauron considered Morgoth to be a failure and set up Morgoth as a false god in Númenor despite knowing full well that Morgoth was powerless and that Eru was the one true God.

That is not a sign of being disloyal at all to Morgoth but to Eru.

Professor Tolkien seems to think that really it was a matter of convenience, not loyalty, but it's still worth musing upon. I suppose Sauron could have invented any phantom for the Númenoreans to worship, but can we imagine, perhaps, that as presumably well-educated people steeped in the lore of the First Age that the Númenoreans were well aware of Sauron's prior affiliation with Morgoth and that made propounding Morgoth, rather than some made-up figure, a more believable lie?

Loyal to himself...who else to be loyal towards? Morgoth was in the void...that he forced the numenoreans to worship Morgoth is another action that suggests he still was loyal.

Personally I think this idea of "bringing Morgoth back" is not only impossible but irrelevant because it has no basis in anything Professor Tolkien wrote, but Sauron could have stayed loyal by carrying on in Morgoth's name. The impression I get is that by and large Sauron did not do so, and when he apparently did do things "in Morgoth's name" it was for the sake of his own power and not in tribute to Morgoth (the establishment of darkness-worshipping religions, for instance). One of Sauron's great lies to Ar-Pharazôn was this concerning the identity of the "Lord of the Darkness":

Yes it's impossible for him to bring him back...
Yeah when he apparently did so it was for his own sake...whose other sake would it be for?? Morgoth was outside the world remember...you almost make it sound that the only way Sauron would be able to stay loyal to Morgoth would be to create a statue of him and start worshipping him and forcing others to do so. That would be stupid and Morgoth would agree. I still hold this view and your post has so far only encouraged me to do so.

Sauron is more loyal to Morgoth than he is to the valar by being loyal to himself.