Hotspot while charging

No, and no.

In fact, it’s a good idea to plug the phone in, as the hotspot uses a lot of energy and will drain the battery pretty fast.

There is no reason why a iPhone can't be connected to power regardless of what it's going. Reducing the amount of battery use is probably the best thing that can be done to reduce the loss of battery health, as racking up cycles is inevitably going to lead to declining battery capacity.