CDEWannaBe

The question you had about hypo/hyperlycemia is called dawn phenomenon.
The diabetes doc was a little off with the timing. Typically the base rate of insulin a person needs drops between Midnight - 3am. Then around 4am the liver starts releasing glucose, which causes blood sugar to rise. This "dawn phenomenon" is thought to give the extra glucose a person needs to wake up and transition out of the parasympathetic (rest and digest) stage.
One of the disadvantages of an insulin like Lantus is it gives one, flat base rate of insulin. In reality our bodies need different base rates at different times of day. That's what contributes to many of the highs and lows diabetics using insulin have.
I'm a type 1 diabetic and use an insulin pump, which uses just short acting insulin like Humalog or Novolog. I can set my base rate to what is needed, which for me is seven different rates between .06 and 1.7 units an hour. I get the highest rate during the early morning dawn phenomenon time and the lowest rate around midnight when my insulin needs drop. These rates keep my fasting blood sugar steady. When I eat I program the pump to give additional insulin to cover the carbohydrates consumed and also to give extra insulin to correct a high blood sugar.
Hope that helps.