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Max rating of the pipe. I didn't check before I sent it but I thought generally the plastic pipe makers were ok upto 80/90C. You wouldn't want to run like that* but they need to be able to stand behind the product if someone does. * Older hospitals, hotels, etc, often have very hot loops and mix down near the consumers.

Hot is only really hot while the water is running. Then it will cool down. Constantly circulating hot loop will always be hot, and given the specs they have to assume the water will be 80C+ continuously.

I thought the way to do this on a new install is to have sensors in the rooms with taps to switch the pump on (with heat based shutoff) when the rooms are occupied/light is on? Doesn't this avoid problems here?

You won't find a variable resistor anywhere near modern power electronics. The electronics switch the power on and off very fast (kHz, not Hz) and the amount of output power can be controlled by varying how long the power is switched on each cycle. Any feasible circuit will monitor both the voltage and current and both are used for feedback. You can go down a big rabbit hole if you want to know how it works. A quick google gave me this, not saying it's the best place to start but this will take you as deep as you like. It's a reference design for a 99% efficient hybrid inverter. https://www.ti.com/lit/ug/tiduf64c/tiduf64c.pdf?ts=1786346438488

By controlling the inverter. To export more current the inverter has to boost the voltage target voltage higher and vice versa.

Weird I would have thought limiting by current would be easier to implement.

Yes, but mostly no. Technically the nominal voltage is 230V but for the vast amount of consumers the voltage at their socket is closer to 240 (likely higher). The limit is set by amps to avoid any confusion here. You can export 16A whatever your local voltage.

It's 16A per phase, ie, 3.8kW

Hang on. You only had the fridge fitted two days ago? Are you sure it's a mouse that damaged the cable, not an installation error? Why not get the installer back to deal with?

Well that's annoying. I did think the price on that one was low and looked in the details to try and confirm it did come with everything. Sorry. Loads of other options there that will come with everything. You don't happen to have an old one of those blocks lying around the house? (I know it's your mums but it's just one of those things that is incredibly common to have stashed somewhere, at least for older people). You absolutely need a proper connection. Don't attempt to jerry rig something. Better to stick with the plug + extension. Can't help any further with fishing the wire. Coat hanger is a good idea, just make sure you aren't poking it anywhere there are live electrics (turn off first if in doubt).

Obviously the fridge was previously installed so it's likely possible with the factory plug on it. However, given you've already chopped that off and are making your own connection: 1. Just use a long cable a feed it through before you make the connection. 2. Then make the connection before you slide the fridge into place.

Agree this is preferable, especially if doing lots of lights like this. In the case of only one or two light strips in the whole house then maybe it makes more sense to keep it local. Yes. Not something I thought about earlier given the electrician apparently planned it this way (and you'd expect them to be aware of requirements).

Why not? Good smoke detectors come with a fairly large base that can be fitted over a larger hole without issue (or over a recessed junction box)

Roughly yes, though from the earlier photo I didn't expect the lead to need to be cut that short. (It wasn't clear where the damage was) I'm not an electrician but I don't think this is a problem. Keep the plug outside the fridge metalwork. If you are going to buy different cable then you can get joint boxes to replace the plug + socket. Will be a neater/more compact job. For example: https://www.screwfix.com/p/vimark-30a-chocbox-connector-box-90-x-30-x-50mm-white/194vt

Can you show us the damage to the cable? If it's tricky to access, maybe you can just cut it shorter and put a plug/joint on the shorter cable?