Is there a comprehensive list of real-number-describers that allude to the properties of that number?
For example:
Amicable numbers, Abundant, Deficient, Perfect, Carmichael, prime, transcendental, etc.
1. If so, where can I find it?
2. What are some new, exciting types of numbers with interesting properties?
3. What is name of the type of irrational number that contains every natural number as a substring of its decimal expansion ?
On
For 3, the general type is a normal number, though this is more restrictive than your request. These will have all sequences present in the proper proportion. The specific one $0.123456789101112131415161718 \ldots$ is known as Champernowne's constant
Wikipedia provides a solid list of numbers: http://en.wikipedia.org/wiki/List_of_numbers