1) "On a cold morning" --- usually "in" is the preposition needed for "in the morning, I..." but because you have used an adjective as a descriptor in the sentence, "on" is needed instead.
2) Firstly I'd like to say that this saying is not so common anymore but to let you know the word "bright" by itself, is still often used to describe someone smart.
3) "grounded" is a punishment given by parents to their children for bad behavior, where they would have to come home directly from school and then are not allowed out to see friends for the period of the grounding.
4) "bounded" means to cover large distances quickly because each step is far apart from the previous one, in an almost jumping like motion but still more like running. It can also refer to a positive energy being when doing so but not always.
5) "Help yourself" or "Drink as much as you want", personally I would use just the beginning or end part because I think using both together is unnecessarily long but is it grammatically accurate, so good job.