1) Your guess is partly true - the demons envy the gods' real power. But the question is more complicated, and there is no good answer to it, because the psyche of demons cannot be fully explored by our mortal minds ;) Demons plot and conspire among themselves, competing for power, knowledge and influence. Probably they would like to be worshipped and have many mortal followers even if it wouldn't give them many tangible benefits, other than satisfying their own vanity. But the ability to recruit and use human agents can be useful for many other reasons. It is also important to remember that demons are different - not everyone is interested in becoming an object of worship.

2) When a being moves to another world, its form is adapted to the new place. In the material world, the demon takes material form, and in the abyss it has an abyssal form :> This is the rule, but perhaps there are some exceptions to it. Demons may differ a little in appearance on subsequent summonings, because the process of taking form is imperfect.

3) You know, it's a bit about perspective. You and I are human, so to us the appearance of the average demon can be disgusting/terrifying (not just the appearance, but the personality as well). Of course, their deformities can also be explained by the imperfection of the materialization process in our world, and that's probably sometimes the point.
Demons are more extreme than humans, they have broader tastes, they are more volatile and crazy, their morals are different from ours, and as immortals they also have a different point of view. Their default appearance reflects these differences. The more subtle and powerful demons are able to influence their appearance in a limited way, so they may want to appear more attractive in the eyes of humans (also to distinguish themselves from the lesser demons).
Nyx is unique in many ways. Maybe her appearance is a coincidence, maybe she just likes this form, or maybe it has something to do with her patron.
Lilith wants to be beautiful and seductive, so that's the form she chose for herself (or that's the way we see her, it's hard to say). And demons like Moloch just don't care about their appearance (or even want to be disgusting).