Yes. Plain water isn't a food-safety risk the way hot leftovers are, but a big pot can still raise the temperature around it for a while.
The old warning about never putting anything hot in the fridge was really about food safety, not water. Water has no proteins or sugars for bacteria to feed on, so the danger-zone rule that applies to soup or rice doesn't apply here the same way.
What actually matters is simple heat transfer: a near-boiling pot warms the air pocket around it, and if the fridge is already full, nearby milk or meat can sit warmer than it should for a short stretch.
Bacteria multiply fastest between 40°F and 140°F (4°C to 60°C), which is why cooked food is supposed to cool before storage. Plain water carries none of that risk, so you're not gambling with spoilage by chilling it while it's still warm.
A single cup or quart barely registers. A full stockpot at near-boiling temperature can push nearby air up several degrees for 20-30 minutes, which matters most in an older, overloaded, or already-struggling fridge.
Thin glass can crack from thermal shock when hot liquid hits a cold shelf. Stick to tempered glass, stainless steel, or plastic rated for both hot and cold use, and leave a little headroom since liquids expand slightly as they cool and contract.
A cup of hot water in the fridge is a non-issue. A full pot is fine too, just split it up first so it cools quickly and doesn't drag down the temperature around your other food.
No — that's different. Hot food should cool for a bit first, ideally split into shallow containers, so it doesn't sit in the bacterial danger zone too long or warm nearby leftovers.
Thin or old glass can crack from the sudden temperature swing. Use tempered glass, stainless steel, or heat-rated plastic instead, especially for anything near boiling.
A small glass cools in under an hour. A full pot can take 2-3 hours unless you split it into smaller containers or pre-cool it in an ice bath first.
An occasional pot of hot water is no problem for a normal fridge. It just has to work a bit harder briefly, the same as after loading in groceries.