The vibrational intangibility was not improvisation, it was a secondary part of the Nuva power which Pohatu had to sort of discover on his own through sensing what the mask could do and theorizing. And the presence of a secondary power like that is mostly unique to the Kakama Nuva, so no, another Toa will not be able to accomplish a comparably extremely different effect with their own masks.
There may be a few exceptions. The Great Mahiki for example adds the secondary power of shapeshifting.
As for improvising, though, within a much more narrow bounds of similarity, it can't be ruled out, but it may be impossible with most powers because the powers are often basically "switch on and aim" type powers. Elemental powers by contrast are highly nuanced in how you can control them, and can be thought of as basically synonymous with improvising.
For example, the Pakari is stated to make the user strong. There's nothing implying that they could focus this power on just specific parts of them. If they could define a zone for the power like that, it would beg the question of why they can't normally extend the power to other targets instead of just the user. However, perhaps the Pakari Nuva does give more nuanced control, since it can extend strength to others.
There also can be some in-between powers. Mahiki's normal power is a good example; the end product will need to be improvised in order to make any image at all, and therefore you have free reign to imagine whatever image you want to project. So that one goes way beyond just switch on and/or aim. And even in the aim category there is more control than the mere "switch on" powers like Pakari. Telekinesis for example gives you variable focus effectiveness, and you can move the object around where you want.
But then again, it may be only an assumption that any of these powers are literally "switch on" and don't give a gradient of options between nothing and fully on. It could be that the Pakari can let you become only partially stronger, but that users normally just amp it up to max every time since that's more likely to be needed.
Re: Kaukau working in other liquids -- there's actually slight evidence for it since we already know it works in two different liquids, although those two are intentionally mimicking each other -- real water and protowater. IMO the power just takes oxygen that is absorbed into a liquid and pulls it inside the mask into air form. (Possibly also the other components of normal air if present, but just oxygen would do.) Therefore, as long as the liquid wasn't inherently harmful to the user, and contained dissolved oxygen like water does, it should work. However, the range of liquids that you're likely to run into with those properties is probably very small.
To sort out poisonous gas? Maybe. The question would have to be if it works on any "fluid" (which can be defined as including gasses, referring to their non-solid state) or only liquids.
One answer to this would rule out working in other liquids -- if it works only by a limited Water elemental power that is telekinetically holding the water outside the mouth, plus a bit of limited Air control to move the air in. If that was the case it could not work in any other fluid, whether liquid or gas.
If it works like a forcefield that pushes out any non-Air, though, then logically it should work in any fluid with dissolved oxygen inside it and that's likely to include virtually all poisonous gasses. (Assuming the gas isn't poisonous to their other biological components or highly acidic.) I don't know of any way to get an exact answer, other than Greg, though. If you're curious enough you could try searching the old S&T to see if he ever answered this. He may have; it seems slightly familiar.
Rau -- Actually there may be room for some improvising in a few situations. Say you're talking to an enemy who has demanded you translate something or die. They'd have no way of verifying what you say, so you could read what it says, deduce why they want this info, and imagine a cover story to speak instead. And the Great version opens up the possibility of talking to a wide variety of Rahi; it would be up to you to think of this, choose which Rahi (if multiple types available) and then what to say, how to convince them to help if necessary, etc. If you're skilled you could potentially enlist a vast army of trained creatures to obey your every command, giving tons of room for improvisation.
Well, the Pakari can strengthen your legs. In BIONICLE: Chronicles 1, Kopaka finds his Pakari and does an extended jump to (try to) make it across lava.
Was it stated to be just his legs? Switching on the whole Pakari power would include the legs, so just from how you worded it here, that doesn't imply it was only the legs. It might be useful, though, if it allocates greater strength to the part you limit it to. I don't recall anything like that being stated but that's my mem...