When Balrog issues are debated one idea which is frequently raised is the possibility that Balrogs were able to change their shapes. If true this would automatically settle many of lingering questions, but the shape-changing ability is itself contested.
There are several passages indicating that the Ainur were able to take on any form they chose. Generally the analogy of 'clothing' is used;
"These were the Valar, and their lesser attendants. ... It was because of their love of Ea, and because of the part they had played in its making, that they wished to, and could, incarnate themselves in visible physical forms, though these were comparable to our clothes (in so far as our clothes are a personal expression) not to our bodies. Their forms were thus expressions of their persons, powers, and loves. They need not be anthropomorphic (Yavanna wife of Aule would, for instance, appear in the form of a great Tree.)"
"Now the Valar took to themselves shape and hue; and because they were drawn into the World by love of the Children of Iluvatar, for whom they hoped, they took shape after that manner which they had beheld in the Vision of Iluvatar, save only in majesty and splendour. Moreover their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being. Therefore the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present."
Given these and the fact that Tolkien eventually came to view Balrogs as Ainur it is sometimes argued that we need look no further, the Balrogs must have been able to change forms. Counter to this is the view that while all Ainur had this ability in the beginning Melkor and Sauron both lost it, and there is some evidence suggesting the Balrogs may have as well;
"Here Pengolodh adds a long note on the use of hroar by the Valar. In brief he says that though in origin a 'self-arraying', it may tend to approach the state of 'incarnation', especially with the lesser members of that order (the Maiar). 'It is said that the longer and the more the same hroa is used, the greater is the bond of habit, and the less do the 'self-arrayed' desire to leave it.'
'Melkor alone of the Great became at last bound to a bodily form; but that was because of the use that he made of this in his purpose to become Lord of the Incarnate, and of the great evils that he did in the visible body. Also he had dissipated his native powers in the control of his agents and servants, so that he became in the end, in himself and without their support, a weakened thing, consumed by hate and unable to restore himself from the state into which he had fallen. Even his visible form he could no longer master, so that its hideousness could not any longer be masked, and it showed forth the evil of his mind. So it was also with even some of his greatest servants, as in these later days we see: they became wedded to the forms of their evil deeds, and if these bodies were taken from them or destroyed, they were nullified, until they had rebuilt a semblance of their former habitations, with which they could continue the evil courses in which they had become fixed.' (Pengolodh here evidently refers to Sauron in particular, from whose arising he fled at last from Middle-earth.)"
VT 39, Osanwe-kenta note 5
While this "evidently refers to Sauron in particular" it is clear that the passage indicates more than one of Melkor's "greatest servants" became bound into "the forms of their evil deeds". It seems reasonable to suppose that some of the Balrogs had suffered this fate since there isn't any clear reason why they would be immune when Sauron and Melkor were not. Especially given that the Maiar were earlier said to be particularly vulnerable to this effect and the Balrogs certainly 'did great evils in the body visible' and seemed likely to be "consumed by hate".
The search must then turn to the question of whether we ever actually see a case of Balrog shape-changing in the texts. While there are no known examples of a Balrog taking a fair form or changing into an animal such as Sauron did to fight Huan there are a few passages which might be read to suggest shape-changing;
"The Balrog made no answer. The fire in it seemed to die, but the darkness grew. It stepped forward slowly on to the bridge, and suddenly it drew itself up to a great height, and its wings were spread from wall to wall..."
FotR, The Bridge of Khazad-Dum
Taken by itself this might be taken to suggest that the Balrog actually 'grew' to tremendous size, and was thus able to change its form. Other readings are that the Balrog was always gigantic and only then clearly seen, or that the appearance of great size was in some sense an illusion or literary device. In support of this last can be found several similar scenes;
"Gandalf's eyes flashed. 'It will be my turn to get angry soon,' he said. 'If you say that again, I shall. Then you will see Gandalf the Grey uncloaked.' He took a step towards the hobbit, and he seemed to grow tall and menacing; his shadow filled the little room."
FotR, A Long Expected Party
"If I had killed the real Strider, I could kill YOU. And I should have killed you already without so much talk. If I was after the Ring, I could have it - NOW! He stood up, and seemed suddenly to grow taller."
And many other examples... Galadriel's temptation by the Ring, Gandalf's healing of Theoden, Frodo's intimidation of Gollum at Emyn Muil and Mount Doom, Sam's encounter with Snaga on the stairs of Cirith Ungol, et cetera. In each case there is a suggestion of growth and other changes in appearance. That even Strider was able to achieve this effect without apparent aid from an item of power (as Frodo and Sam may have had from the Ring when they seemed to change) might suggest that it is an impression rather than an actual enlargement. In many cases perhaps an 'illusion' of great size due to the power of the being or that which they bore.
Tolkien even supported this possibility in one of his notes;
"There is a pencilled note written on the manuscript against the description of the Balrog: 'Alter description of Balrog. It seemed to be of man's shape, but its form could not be plainly discerned. It FELT larger than it looked.'"
ToI, The Bridge
Still, there are other possible examples;
"His fire was quenched, but now he [the Balrog] was a thing of slime, stronger than a strangling snake."
TT, The White Rider
"Far beneath the ruined halls of Angband, in vaults to which the Valar in the haste of their assault had not descended, Balrogs lurked still, awaiting ever the return of their Lord; and now swiftly they arose, and passing over Hithlum they came to Lammoth as a tempest of fire."
Silm, Of the Flight of the Noldor
With these two passages the question becomes whether these features ('a thing of slime', 'a tempest of fire') were figurative descriptions, and if not whether they necessarily would require a change of form. With its fire quenched from the fall into water the shadowy power of the Balrog might well make it seem a 'thing of slime' without any need of shape-changing. Likewise the 'tempest of fire' could equally well be a figurative or literal description of the flaming auras of the Balrogs rather than an indication that they had transformed into an ACTUAL firestorm.
In conclusion, while there seems every reason to believe that in their origin the Balrogs would have had the shape-changing abilities common to all the Ainur it is unclear whether or to what extent they retained these abilities in later years. As the "bond of habit" grew greater "the longer and the more the same hroa is used" it seems likely that after frequently appearing in the same shape throughout the Elder Days and the Three Ages of the Sun the Balrogs would have become somewhat tied to it, but whether and how much this would limit their ability to appear differently is unclear.