Unplug laptop while it is running

Nope, it should be fine. Laptops are designed to switch between battery and mains power.

Stuff to watch out for? Tripping hazards. While barrel connectors are fairly robust, they've been known to fail — especially with a sideways force. Unplugging the power connector totally would mitigate both this and tripping risk. While there are special mechanisms for laptop HDDs that park the head, this is in case you drop it.

In essence, anything that can kill a laptop while moving it would kill it anyway. I've had a few desktop divas experience the same failure modes, so... it's not especially dangerous to move a laptop.

Your brother-in-law has an outdated view of how rechargeable batteries work. Older laptops used NiCd batteries were susceptible to the memory effect. Their maximum charge could be reduced if they were repeatedly partially discharged and then charged. There were all sorts of attempts to mitigate this, including waiting until the battery was discharged before charging it again. It's debatable whether the memory effect was real.

Modern laptops use lithium-ion batteries which have no such problem. They also have sophisticated hardware and software to monitor the battery, keep it in good condition, and prevent anything a consumer is likely to do from harming it.

Unless your laptop doesn't have a battery, you're fine. Leaving it plugged in all day, everyday can minutely reduce your battery's efficiency. But since your battery degrades overtime anyway, that's not really an issue. Lithium batteries don't last forever, nor retain their peak efficiency indefinitely.