Humans give off heat at fairly consistant rates, if this energy is not diverted somewhere it accumulates, also you cant shut down all power sources as air still needs to be circulated and refreshed along with lighting, sensors, computer systems etc so the ship will always be generating some heat on top of that created by the crew so yes they will fry if you dont get rid of the heat
Dumping any kind of heat when trying to be stealthy will make you easier to detect.
With good enough insulation, this can indeed hide you for a while. For how long, depends on the insulation and on whatever you're using to store heat. You can't stay hidden forever, but nobody says you need to. All you need is to get to a good hiding place where you can dump heat unnoticed (like behind an asteroid, or in atmosphere.
How do you plan on getting to that hiding place when your engines are offline, and the enemy vessel has your last known location and last known trajectory recorded? All you are doing is stalling for time, and hoping that your enemy will think you are too stupid to actually try stealth in space.
If you have the missile doing random zig-zags the last few light-seconds to the target and detonate at a couple tens of thousands of kilometers, it would require a statistical miracle for point defenses to shoot your missile down.
This is assuming a lightspeed cap on travel, sensors, and communications...
There's a little something called inertia that can and will keep ships and missiles from doing zigzags in space. A missile or ship that is moving at high speed will take some time to slow down or change trajectory.
Also, if the target ship is moving slowly relative to the missile, the target ship might be able to manage a major trajectory change, resulting in the missile not being able to correct its trajectory in time and overshooting its intended target.
Additionally, missile tracking systems just won't be as sophisticated as a starship sensor package (at least not without making the missile really expensive). A missile could actually be spoofed by decoys and ECM. Those same anti-missile systems won't spoof a sophisticated ship sensor package, but at least the missile didn't hit.
If I do have an FTL drive, I could always use that as a last-resort missile dodging technique. All I need to do is jump a relatively short distance and all the missiles will be on a completely incorrect trajectory to hit me.