How to Define and Implement (Single-Writer) K-Atomic Snapshot Object?
Regarding Problem 1: I think the first part of the definition is fine. 
The second item (about the time before $scan$) is not clear to me; More specifically, the order of quatifiers: is it $\exists time: \forall i$ or $\forall i: \exists time$ ? I guess the first, because in the second I don't see why it is necessary. But definitions should not need guessing ;-)

Regarding Problem 2: In principle there is two ways of looking at $K$. You can consider the last $K$ updates on a per item level or the last $K$ updates globally. The other choice is (already mentioned above) whether to require a point in time when the memory actually had the value returned by $scan$ (the $\exists time: \forall i$ case above) or whether not to require such a point.

A general comment: What happens when every process just updates $K$ times and scans until two scans are equal. Wouldn't that (trivially) implement ($1$-)atomic snapshot memory? If that is the case then the whole idea of $K$-atomic snapshpot memory could be hard to sell.