What is a “double” in dynamo and what does mean? I remember reading them somwhere, but I forgot. is double just a number?
Double means a decimal numerical value - so a number with decimal places e.g 2165.2654. An integer is a number with no decimal places e.g. 2165.
 means an array of arrays or lists of lists. So the values are within lists of that Type of value e.g. int is a list of integers. You can get var which means that the list of lists holds different Types of values e.g. string (text), int (integer) double (decimal numbers) etc. Sometimes lists are simply declared as the var type as they are not explicitly set to take just one variable type.
Hope this helps!
Just for reference, here is a table of numerical data types in a C# environment.
Then why not call it rounding? Double makes ZERO sense
This is also what the Integer (number) node is for
How do you apply this to a vertical height input then?
Double is a programming term and makes perfect sense. You can read about why it is called a Double here.
In your example, this would most likely be a double as 10.0, Revit typically stores dimensions (and most other numerical values) as doubles.