[A_Brandybuck]

I think, that the Ainur themselves have some restrictions, when they are embodied.
In the case of Sauron being hold by Huan, we see that Sauron could not change his shape in the way, that he can escape. He can change his body, but in exact the same location, so that Huan can hold him still. Not until his body was destroyed (or dead), the mere spirit can go, wherever he wants.
What then happens, we can see it in the Fall of Sauron in the War of the Last Alliance.
Sauron could be slayn, and he was slayn by Gil-galad and Elendil. His body was dead and his naked spirit escaped, so that Isildur could cut the Ring from his finger. The Ring is not a part of the spirit of Sauron.
Sauron needed thousands of years to reembody himself totally, without the Ring his power was diminished, that could be the reason, why he took so long. After the Fall of Numenor he needed not so long, the Ring was there in Barad-dur.
Another case, from which we can conclude some details is the case of his holding in the 'hands' of Huan. Sauron has a great fear losing his body and going naked to Melkor. I conclude herefrom, that Sauron could not reembody himself directly, because Luthien said, he shall endure the torments of his master everlastingly. 'Everlastingly' seems to me a little bit exaggerated, we know, that he could reembody himself at any time.
We can extract from all above, that in spite of the possibility of changing the shape, an embodied Maia could not leave the body, when he is hold by someone. What exactly happens, when a Maia leaves his body, I don't know, because there is no quote concerning this. He can change his shape, but he cannot escape through the mere changing.
When the body dies, the spirit can leave the body, but cannot reembody itself directly. Reembodying is apparently dependent how much power the spirit has.

Coming to the different shape a Maia (representive for Ainur) can have. Maia have a gender. I don't know a case, where an Ainu choose a body of the other gender. Apparently there is a restriction, that a Maia can embody himself only with shapes of his own gender. The gender is anchored in the spirit of the Maia and influenced apparently the physical form. An attribute. But surely there are other attributes, which are fixed in the spirit. Attributes, which have effects of the physical forms they can choose. I think, that the body of the Maia is mirroring his spirit. We see this in the changing of Sauron's shapes in the 'hands' of Huan.
The physical form of a Maia is consquently a vague image of the spirit.
But it is also possible, that the attributes change, so that a Maia can (must) take other shapes as earlier. We see this in Morgoth and Sauron. Both had a beautiful shape, but with doing continously evil deeds, they are shaped in dark form.
After the Downfall of Numenor, Sauron could not take a beautiful shape again, he lost his shape in the Sinking of Numenor. I interpret this passage in the following way: the Downfall of Numenor, for which Sauron is responsible, the measure of his evil deeds was so high, that 'his spirit' was so foul, that he could not take another form as a 'Dark Lord'.
Some may think, that he did very many deeds already earlier, why he lost the possibility of a beautiful shape so late. There are two possibilities:

#1 His body died, so that his new body cannot have a beautiful form. The new attributes can only be implemented after a maltreatment of the body. The thesis could be supported by the fact, that Melkor's hands are for forever black, after his hands were burned by the Silmaril. In both cases, the maltreatment is the bill for the evil deeds.

#2 The measure his very high. Melkor could live in Valinor in his beautiful shape, although he has many evil deeds behind him, including his most evil deed: the creation of the race of Orcs (supposed they came from Elves).

Maybe it is a combination of both.

I come to the Balrogs. Balrogs are Maiar, which came into the service of Melkor. They became demons of fire. Apparently they could not change their body, nor leave it. The one Balrog, of which we know, that he lives in the Third Age, has not changed his form after that really long time. I think that these Maiar are bound to the flesh. And there is the connection to something over what, we know more: the Istari.
It is (at least) for the Valar possible to bound a Maia to a body and to enjoin them the restrictions of the flesh. We know, that the power of the Istari was also diminshed, because they wanted only a passive supporting of the free peoples of Middle-earth against Sauron.
My theory is, that if it has been possible for the Valar to bound a spirit to the flesh, it would be also possible for Melkor to bound the Valaraukar to the flesh. But in this case the power of them wasn't diminshed, maybe increased through subcreation from Melkor.
Melkor used them as warriors. The slaying of a Balrog would be the escaping of the spirit from the body, but without chance to rebuild the body or getting again the power (only through the creator). I conclude this through a comment of Gandalf (I don't find it at the moment), that Saruman is nothing more like Sauron. A mere spirit in the wilderness.

Looking now to the chain, which Melkor enchained, I would say, that it is possible to forge a chain, which can hold the physical form of Melkor enchained. Huan get it with Sauron, why could not Aule forge a chain, which is mighty enough to hold Melkor. The chain must only resist the changing of a shape (that this is possible, we see in Huan) and resist the destroying through the power of Melkor (he lost much with subcreation). We know, that leaving the physical form is not possible, when the person is in a form enchained.