If you have a nuclear reaction that is going out of control, then you have to get it in control. Shutting the plant down would mean you don't have the ability to use things like the control rods to do this.

No, the control rods are constantly forced into the core by passive systems (hydraulic pressure, gravity, springs), and only stay withdrawn because of active systems. If the active systems lose power, a few seconds later the control rods will be fully inserted. Many reactors also have pre-pressurized tanks of neutron absorbing fluid connected to the core, a sort of liquid control rod. Provided one of the redundant valves can be opened, that will also quench the nuclear chain reaction. (And I wouldn't be surprised to learn that the valves have a ratchet that makes them stick open until some poor bastard visits them in person with a special tool.)

It's also worth pointing out that many safety systems have no self-protection features like circuit breakers, or even off switches where a well-meaning idiot might turn them off just because fire is shooting out. If a back-up cooling pump develops a short circuit or a bad bearing, it will continue to run until it destroys itself. The idea is that the protection equipment will cheerfully use itself up to protect the main plant.

The problem here is the forensic technicians. Every single one of them needs to be fired.

Indeed. With few exceptions, every scientific instrument that uses fluids is wildly inaccurate. (Precise as hell, but inaccurate.) That means pH meters, gas chromatographs, mass spectrometers, biomolecule sequencers, you name it. To get reliable measurements, you have to run an end-to-end calibration at least every few test runs. For ultra-trace detection, like DNA and high-potency drugs, often every single sample needs to be calibrated.

Failure to do this on such an epic level draws all results of all the labs into question. Even in an academic lab, getting caught at this would be a career limiting move.

This case is more like, if I ask you a question, you provide an answer, and then later I ask you the same question. You can't claim the 5th this time around, because you've already provided the answer.

Yes, you can. The 5th Amendment does not protect privacy or secrecy, it controls the actions that the government is allowed to take. The government can never compel incriminating testimony. Period. You can carve a murder admission onto a bailiff with an icepick one day, and then the next day refuse to repeat the statements and the government cannot compel you to.

If this were not the case, then any statement that merely suggested guilt would justify torture to extract all guilts. This is not melodrama. The Puritan exiles to America made this rule to prevent the system they were fleeing, a system where a minor infraction would be noted, and the defendant would be dragged before a court and asked to take an oath of total innocence. If they took the oath, the infraction was brought out and they, being guilty, were tortured to extract proof of all other guilts. Thanks to them, incriminating yourself does not grant the government one iota of power to extract further incrimination.

"No person ... shall be compelled in any criminal case to be a witness against himself ..."

The I-35 bridge collapsed because of a design flaw. (Mostly. The inspection engineers had a case of tunnel vision.)

You mean besides the Emergency Alert System, which is required by law to be supported by a wide variety of radio transmitters.

One of the more mind blowing things I read in 2008 was the discovery of a third type of visual receptor besides rods and cones. Essentially there's a third type of receptor that only detects sort of gross levels of light, and feeds directly into the system which regulates your circadian rhythm and is used for some other purposes.

Those are the photosensitive retinal ganglion cells. In addition to driving the circadian rhythm generators, they also control pupil size in response to light. IIRC, research in cats found that they do connect to the visual cortex, although how the signals are perceived is not yet known.