Dembski's paper shows that any search performing better than random sampling requires the addition of information. This is information in the Shannon sense, -log2(p), where p is the probability of sampling the target region in a search space normalized according to random sampling.
This cannot be done algorithmically for an arbitrary search landscape. According to IDT, however, the defining quality of intelligent agents is their ability to produce information. This can be measured and thus falsified or confirmed. I.e. do agents we normally identify as intelligent, such as humans, actually produce information?
Per question 2:
If the above does not define terms enough for you, let me know. Dembski does a very good job in defining ID mathematically. The following is a collection of his work, much of it freely available online.
How does it show this? And for what definition of "information"?
it is mathematically provable...
The first step of mathematics is to define your axioms and then your terms. IDT doesn't seem to do either.