What is the standard operational battery temperature for the Note 9? Bought the phone a week ago &amp; today it seems to be running slightly warm just surfing sites in Chrome. Downloaded a battery app and see that the battery temp is at 32 degrees Celsius constantly right now for the last 2 hrs. Is this normal?
OEMs hardly provide such data to decide if it is normal or not. Not surprisingly, user manual of your device is quiet on this point. 

This post on SE. Electronics (talks of Samsung), puts maximum safe temperature as 45°C. 

However, temperature has a bearing on the battery performance. See this for details. At 32°C it results in 80% of optimum battery capacity. 

I would consider based on my observations, that this perfectly fine, though Samsung phones have been reported to have  fast charging problems with a low temperature error, which doesn't seem to be your problem.