The regulator should have thermal limiting (check the data sheet for details), and will shut down if the temperature gets too high. It won't damage any other parts, and definitely won't melt any solder!
Using a larger heatsink will reduce the temperature, of course. A switching regulator will be a lot more efficient and will just get slightly warm.
Frankly, if you "don't know where to start", get someone that does. It sounds like the cost of failure is very high, so this isn't the time to experiment and learn from mistakes. Find someone that knows this stuff. You can look over their shoulder and try to do as much of the grunt work as possible so next time you're the expert. I think I underestand the basic principles pretty well and could get something working. However, for extra high reliability there is usually much lore and experience that you wouldn't guess from the physics alone. I wouldn't take this on myself without a expert to at least ask occasional questions of and to bless the ultimate design. So from a non-expert view, here are some thoughts:
First 45C isn't all that hot. It's only 113F, which is not out of line for summer shade air temperature in various parts of the world. Ordinary electronics gadgets, unless specifically for lab use or other environmental constraints, will have been designed to run at 45C ambient. Think about it. If you left your cell phone on the dashboard of your car parked in the summer sun in Phoenix, you probably wouldn't blame it for not working. Of course you probably couldn't hold it without hurting yourself either then. But, if you were standing outside under the shade of a tree mid afternoon in summer in Phoenix, you'd probably be pissed of if your cell phone didn't work. That's 113F.
Second, two other advantages you have is that this 45C is highly reliable, and essentially infinite. In other words, it's 45C all the time whether its summer or winter on the surface. Also, you can dump a few kW into a reasonable size "room" of this mine, and it will still be 45C.
So the best solution, as Russell also pointed out, is to spec things so that they run fine at 45C ambient. Space them out enough so each thing sees the 45C ambient, perferably without needing any forced air cooling. That way you don't have to worry about what happens when the coiling breaks down. Again, most off the shelf electronic gear should already be fine with that. Lifetime may go down, so derate. Get good quality stuff that should last 10s of years normally.
If you really need cooling, my first knee jerk reaction is to build a small room and put 2 or 3 air conditioners in it. Make sure any one air conditioner has the capacity to cool the room sufficiently. The multiple air conditioners are for redundancy. You seem to say it's only a few kW, which is small as airconditioners go. 1 kW is only 3400 BTU/hour. Also, it sounds like every last item doesn't need to be cooled, so some of the power you need can be dissipated outside the room and doesn't add to the air conditioner load. 113F isn't that hot and there are certainly off the shelf air conditioners intended to work under those conditions. Having 2 or 3 units always on cooling the room is for reliability. However, this is a area you really should consult an expert on.
Best Answer
Virtually all manufacturers data sheets will list both the minimum and maximum storage and operating temperatures for their parts. Further many parts will be available in multiple versions designed to operate in more strenuous environments.
So for example an integrated circuit (IC)may be available in the following versions:
Commercial 0-70°C. Industrial -40 to +85°C Military -55 to +125°C.
Needless to say, the cost goes up with increased performance. There are even chips speced for use in outer space. So for example a chip which costs $2 in the "Commercial" grade might cost $2,000 for the version rated for outer space! (As you can guess, they don't sell very many of that grade.)
As to your mouse, generally they try to use the cheapest components possible, so for your mouse 70°C might be the upper limit (it might even be quite less!) which is about 158°F.
One needs to keep in mind that the temperature of the components is affected by two sources:
1) The temperature of the environment. 2) Self heating - some components generate a large amount of heat in operation, CPUs in PCs are one example. This is why they are fitted with big heat sinks and cooling fans to keep them from getting too hot.
As a general rule of thumb, the more power a chip consumes the more heat it generates and the hotter it will get. Of course a mouse uses hardly any power so this should not be a big deal for a mouse.