Is it normal for the lower radiator hose to always be cool or at room temperature even when the car is totally warmed up?

That's the goal of the cooling system. Coolant takes heat from the engine, sends it to the radiator where that heat is given up to the air flowing through the cooling fins. Once that heat is drawn off, the cool coolant flows back into the engine to absorb more heat and start the process all over. Sounds like your cooling system is working fine.

If you live in an area like I do where they throw a pound of salt on an ounce of snow, it is common for those cooling fins on the radiator to rot and crumble away. That makes it harder and harder for the heat to get moved into the passing air. That's when the coolant going back into the engine will be too hot and overheating will start to occur. The symptoms will be a little different from what we normally see as sudden catastrophic overheating. Instead, you're likely to see the temperature gauge run a little higher than normal at first, and the amount of overheating will be worse on hot days, and barely noticeable on cold winter days.

Here's links to some articles that will do a much better job than I can of explaining cooling system operation:

https://www.2carpros.com/articles/engine-cooling-system

https://www.2carpros.com/articles/how-a-radiator-works

Besides removing heat, the cooling system is also responsible for getting the engine up to normal operating temperature as quickly as possible. That is done in large part for lowest emissions, but also because internal parts expand and fit best at the desired temperature, typically around 195 degrees F. Pistons in particular are oval-shaped when cold. They expand more in one dimension and become nice and round and fit properly when up to the correct temperature. When cold is also when most engine wear takes place. Very little wear occurs once it's warmed up.