Loosely the main points are that the rate falls off exponentially with the "width" of the barrier, and that energy is still conserved (you can say the particle "must have" had more energy at some point because it crossed the barrier, but you'll never observe it with more energy than it has). For more precise answers, just plug the potential you're trying to tunnel through into the Schrödinger equation.
You're referring to the 'Gamow window' right? From what I understand, the Gamow window only refers to the energy levels of the particles, rather than temperature per se. I'm guessing that's why approaches like Muon-catalysed fusion can work at lower temperatures.
Muon-catalysed fusion reduces the width of the barrier - the muon is ~200 times closer to the nucleus (since it's that much heavier) and so the coulomb barrier is that much narrower - and the reaction rate is exponential in that.
Sure, it makes sense that it affects the Coulomb barrier (or the Gamow factor/window https://en.m.wikipedia.org/wiki/Gamow_window ), but doesn't muon-catalysed fusion kind of prove that cold fusion could exist? The Wikipedia article suggests that the reactions can take place at room temperature, with the main issue being the reliability of the required reactions.
In other words, we already have one proven method for reducing the Coulomb barrier, why is that the only one that's possible?
The standard model is very solidly established, and it tells us what fundamental particles there are. The muon is the most stable negatively charged particle other than the electron; the Xi is about 10000x shorter-lived, and the tau 1000x shorter than that - and even muons aren't stable enough to be practical.
More importantly, these things are physics rather than chemistry. No amount of chemistry will produce muons from ordinary chemicals, you have to use high-energy radiation. (Admittedly molecular bonds distort electron orbitals a little bit, but nothing like the factor of 200 you get from switching them for muons). If someone had a way to catalyse any kind of nuclear process with chemistry (something as simple as making a radioactive element decay faster) that would be a major breakthrough. To jump straight to the single most useful one would be too much to hope for.
The atoms of palladium in the cold fusion experiments don't know that they're supposed to be doing "chemistry" rather than "physics". That division is only in our theories, not in reality.
> (Admittedly molecular bonds distort electron orbitals a little bit, but nothing like the factor of 200 you get from switching them for muons)
This is a valid concern, but as I understand it, the suggestion is that strong electric or magnetic fields have to be applied in order to induce cold fusion, so we're not just talking about the distortion of orbitals from molecular bonds, we're talking about the distortion due to strong externally applied fields, which can be much larger.
It probably isn't. In fact it almost definitely isn't (there was some work which showed palladium matrices should accelerate beta-decay processes a little).
But there's no model which supports any of the proposed mechanisms for cold fusion currently put forward as possibly workable devices - i.e. it's not theoretically possible, and there's no solid path to show that it might be.
