Why can't a Romulan ship cloak and fire phasers or raise shields while cloaked?
Firing phasers or using shiny things such as shields or warp engines will create a very visible effect one way or another - and thus expose your position even if you yourself remain invisible. But only if your opponent is a starship, capable of sensing things like nadion radiation from phaser preheat or whatnot, and capable of plotting your position and course from such data. Better to drop the supposedly power-consuming cloak, then, and divert that power to tactically useful things.
Remaining invisible in a fistfight does not carry such a penalty. Your opponent can't track you even if he can observe where your punches fall...
It could be that Gem'Hadar can only cloak while standing totally still, because their skin needs a few seconds to adjust to bend light exactly right to make them invisible.
I don't really think chameleon skin has anything to do with Jem'Hadar invisibility. These folks don't just become invisible, they also make the hardware they carry (guns, clothes, sensors) invisible as well.
Probably it's psychic: a Jem'Hadar, when concentrating intensely, can make people around him ignore
him, fail to see him even when he's perfectly visible. Such ignorance will extend to things being carried or held by the Jem'Hadar, too, up to a certain limit. And the ignorance will affect the minds, not the senses, of the opponent - so it doesn't matter if the opponent uses a sensor such as a tricorder, as he will still be fooled into ignoring the Jem'Hadar.
This model covers most of the bases, and is very similar to how certain "shapeshifters" such as the Salt Vampire seem to operate: through illusions rather than through actual if weird physics. There's one pretty obvious hole there, though. Shouldn't opponents located far away (say, on an orbiting starship) have an easy time seeing the Jem'Hadar through their sensors?
Then again, we don't get too many plotlines where this would matter. And we do notice that the Jem'Hadar invisibility doesn't work in ship-to-ship fighting, which we might take to indicate that it's indeed psychic and suffers from range limitations.
Perhaps if the Jem'Hadar concentrate really
hard, they can cloak their ship - which is of no use in combat where concentration of that sort is impossible, but might explain a few feats of infiltration, or the fact that a fleet of hundreds was invisible to the Romulan Tal'Shiar crews, experts in cloaked warfare, in "The Die is Cast"...