Why is the current temperature sometimes lower than today's low temperature? For example right now it 13F with a predicated low today of 16F!

It's a prediction. Predictions aren't always 100% accurate.

Science is a lie!
But they know that the prediction is wrong. So why not update it?
- a) Forecasts are very resource-intensive, they are performed on a specific schedule using a computational forecast model. Updating the predictions would require inputting new data and running the model again, and by the time they do that, the next forecast will already be out.
  
  b) Do they know it's wrong? Where did you get the temperature? From an official weather station? If not, there is no reason to imagine that someone is noticing that this one particular model run was wrong in one particular spot across the whole country and trying to fix it in real time.
  
  c) If you did get the current temperature from an official weather station, that IS your update for it. Real time data from official weather stations is always going to trump the forecast model. What would be the point of updating the forecast when the current measured data from the weather station is now available? That's like driving down the highway and saying "I was predicting my speed would be close to 65mph, but due to the heavy traffic I'm seeing today, I'm going to re-estimate my speed to be 45mph" when you have a perfectly accurate speedometer right in front of you telling you exactly what speed you are going at all times. Forecasts are only useful for the future, and they can be wrong.

My understanding is that the predicted low is for the upcoming night, not the lowest temperature in that 24 hour block

It's right in your title:

predicted low

Predictions are not guaranteed to occur, thats why they call it a prediction and not a prophecy.

true but shouldn't the low be updated with current observations?
- Not when it's early morning. The temperature is on the rise from the previous night. The prediction you're seeing is for the coming night.
- My regional temp estimates are far enough away for a few degrees difference as I am closer to a large body of water. In my case it is only a couple of miles/few kilometers distance for that variation.

The announced numbers are based on the station that records the data which likely has a different microclimate.

An example, the weather station I use is recorded at an international airport so the heat of the sun reflecting off the concrete causes the low to be higher than my place on the other side of a hill which gets the sun later in the day.

Forecasts are generalisation of weather as predicting the future is not possible.

That makes sense since my station is probably an international airport also.

Aren't the highs / lows what were forecast and the current temperature based on actual measurements from weather stations?

I guess they don't update the high/low to match observed readings.

This depends on the implementation of whatever you are looking at. The weather app I use does integrate actual measurements in the forecast for today.

Weather predictions are not 100% accurate.
The place making the predictions may be using different instrumentation or be in a different location than the instrumentation used to give you real time info. This is also why what the app says it is and what a thermometer right in front of you says can also be different.

High is the warmest it will get today. Low is the temperature at sunrise tomorrow morning.

What's that "F" behind the number? Does it stand for "F"uckinghelliamlivinginacountrywithouthealthcare?

You don't know that. I could live in Belize or Montserrat or even the Turkish Republic of Northern Cypress...
- Wait, Belize uses Fahrenheit? I can't Belize it!
What a silly thing to get hung up over.