There's an interesting bit on this topic in History of Middle-earth which comes up in an odd context: the second marriage of Finwë1.
To speak of the dooms of Mandos: these are of three kinds. He utters the decisions of Manwë, or of the Valar in conclave, which become binding upon all, even the Valar, when they are so declared; for which ...
Tolkien tells us that the power of the Ring is Sauron's power.
Now the Elves made many rings; but secretly Sauron
made One Ring to rule all the others, and their power was
bound up with it, to be subject wholly to it and to last only so
long as it too should last. And much of the strength and will
of Sauron passed into that One Ring; for the power ...
Yes. Huan is a Maia.
But true ‘rational’ creatures, ‘speaking peoples', are all of human / ‘humanoid’ form. Only the Valar and Maiar are intelligences that can assume forms of Arda at will. Huan and Sorontar could be Maiar — emissaries of Manwë.
(HoME X, Myths Transformed)
Following the footnote, we find:
See p. 138. - At the bottom of the page bearing ...
They wouldn't be tempted in the first place. All the Valar have decided to abandon Middle Earth in its entirety to live in the Undying Lands, while the Ring tempts people with visions of power over others. Note that Galadriel associates going West with resisting the Ring's temptation:
“I pass the test”, she said. “I will diminish, and go into the West and ...
I think the issue is that you're in part confusing Mandos's decisions with his reporting, so to speak, and then confusing the consequences of what he says will happen.
Mandos is the doomsman of the Valar, but by-and-large, he pronounces what he knows is is fate, not rendering his own verdict.
Námo the elder dwells in Mandos, which is westward in Valinor. He ...
The answer is yes, the host of Men was truly powerful enough that the Valar gave up their custody of the entire world and appealed to Eru.
Faced by this rebellion, of appalling folly and blasphemy, and also real peril (since the Numenoreans directed by Sauron could have wrought ruin in Valinor itself), the Valar lay down their delegated power and appeal ...
Tolkien addresses this somewhat in an essay titled "Notes on Motives" (emphasis mine):
Morgoth lost (or exchanged, or transmuted) the greater part of his original 'angelic' powers, of mind and spirit, while gaining a terrible grip upon the physical world. For this reason he had to be fought, mainly by physical force, and enormous material ruin was ...
Tulkas did fight with Melkor alone in the War of Powers and was victorious. Sure, that was somewhat later, but still before Melkor's captivity.
I think that voids arguments along the lines that Tulkas was merely "the straw that broke the camel's back".
Frankly, I disagree with the above reply. I think all the mentioning of Melkor being the mightiest Vala at ...
I would argue that while Melkor is indeed more powerful than Tulkas, we are speaking of both (overall) power vs. individual strength and focus. Melkor could contest all other Valar because his only purpose was defeating them in order to become the king of Arda (while the Valar had trouble fulfilling the ideas given to them by Eru) and because of the ...
They do have limitations.
The biggest limitation is that they cannot create independent sentient life; as Ilúvatar says to Aulë when he creates the Dwarves:
'Why hast thou done this? Why dost thou attempt a thing which thou knowest is beyond thy power and thy authority? For thou hast from me as a gift thy own bring only, and no more; and therefore the ...
In general, only elves were reincarnated in Mandos
Reincarnation occurring from the halls of Mandos seems to be a thing only for elves, and the single exceptional case of Beren (exceptions like this actually have to be effected personally by God/Eru).
The spirits of elves and Ainur were "immortal"
The Ainur are at a fundamental level immortal, in ...
Did Mandos make any decisions that didn't prove to be disastrous?
Yes, his decision to allow Beren and Luthien back into the world was instrumental in regaining the Silmaril that was set in Nauglamir (the Necklace of the Dwarves), so that it could eventually pass on to Earendil and guide him to Valinor.
See chapters 22 and 24 of the Silmarillion.
Adding onto Shamshiel's answer, Tolkien discusses this in more detail in a letter to family friend Robert Murray, who proofread several drafts of The Lord of the Rings (emphasis mine):
[Sauron] finally induces Arpharazôn, frightened by the approach of old age, to make the greatest of all armadas, and go up with war against the Blessed Realm itself, and ...
No. The Valar were looking out for Men as best they could, but they were hesitant to get very involved, because of the mortal nature of Men and their prior experience with Elves. Remember that Men very quickly fell from grace, as well, so they were in a 'state' more like the rebellious Noldor than they were, say, the Vanyar.
To Hildórien there came no ...
Depending on exactly what you mean by "rules", the answer is probably either "not actually possible" or "bad bad bad very bad things."
Can Gandalf shed his physical form?
This seems to be the assumption underpinning the question; if Gandalf could just dump his meatsuit and fly over to Mount Doom, the story should be over inside an hour, right?
Reasons I could see:
A war with the Númenoreans in Aman would have laid waist to the land, and killed many of the peaceful elves living there. And the Ainur using their physical power is something really destructive for lands, as we know from the fate of Beleriand. That said, I doubt the Valar and Maiar themselves would have suffered much from such a war, ...
He was not alone
It is told among the wise that the First War began before Arda was
full-shaped, and ere yet there was any thing that
grew or walked upon earth; and for long Melkor had the upper hand. But
in the midst of the war a spirit of great
strength and hardihood came to the aid of the Valar, hearing in the
far heaven that there was battle in the ...
In principle, we can assume that any Maiar that can be "killed" is stuck in their particular form; otherwise, what would be the point in destroying their bodies?
This makes things somewhat easier, because those Maiar who have been killed is a relatively short list.
The big ones, of course. A possible reason for their being stuck in ...
No, they are not vulnerable to The One Ring
I'll approach this answer from 2 sides.
Why they wouldn't be tempted by the Ring
Why the Ring would never be in question
Finally, I'll add in some additional information.
Why would the Valar not be tempted by the Ring?
The Valar were the greatest power after Eru himself. They had no need for the One Ring. The ...
No they could not defeat the Valar. Sauron's plan was to convince the king that he could and should defeat the Valar. Sauron knew what the Númenorean forces looked like, and he would not have continued with his plan if he thought men could beat the Valar. The entire point was for the Valar to destroy them. This is another example of why Sauron is called the ...
Some of them definitely did:
In Ainulindalë, Ilúvatar mentions Melkor to Ulmo; if they didn't know of each other before, they certainly did at that point:
Ilúvatar spoke to Ulmo, and said: 'Seest thou not how here in this little realm in the Deeps of Time Melkor hath made war upon thy province? He hath bethought him of bitter cold immoderate, and yet ...
It is not specifically stated that the Trees used the light of the Lamps.
The Valar Yavanna sung them into existence, their light source was their own, with no mention of usage of the lamps existing light.
And when Valinor was full-wrought and the mansions of the Valar were established, in the midst of the plain beyond the mountains they built their city,...
I think you've mostly answered your own question here. The only example I can think of to suggest that they might have been able to win is Fingolfin's ability to wound Melkor:
...and he wounded Morgoth with seven wounds, and seven times Morgoth gave a cry of anguish
Yet with his last and desperate stroke Fingolfin hewed the foot with Ringil
As tall as they wanted to be. Some of the most famous illustrators of Tolkien's work routinely depict Valar as being extremely large:
Ulmo and Tuor, by John Howe
Morgoth and Fingolfin, by John Howe
Mandos and Lúthien, by Ted Naismith
The Valar are archangels. They can do whatever they want. They could be an inch tall if they chose, or they ...
"whereas all the other Ainur who aren't Maiar are identified as Valar." - the Ainur are the "angelic" beings who existed with Iluvatar before the making of the World (Arda). The Valar are those of the Ainur (but not all of them) who descended to Arda and assumed physical raiment, to be Arda's guardians. Melkor was one of them, so techically, he's both an ...