security amplification
As D.W. said, your questions are not very clear. I interpret your first question as asking how the security of an $\epsilon$-PRP varies with $\epsilon$. The answer to this is quite clear from Tessaro's introduction: $\epsilon=0$ corresponds to the standard definition of PRP (also called fully-secure PRP in the paper). As $\epsilon$ grows, the security weakens. If you push to $\epsilon=1$ (which is not permited by the definition), there is no security left.

So the highlighted sentence in your question means that the assumption that AES is very weakly secure (0.99-PRP) is much weaker (and thus more likely) than the assumption that AES is fully secure.

Concerning your second question: The problem solved in the paper is to show that a cascade of $\epsilon$-PRPs with independent keys becomes more and more secure as the number of ciphers in the cascade grows. For $\epsilon\geq 1/2$, this was unknown from previous papers.