Terminology in combinatorics

This is only an answer to your first question.

The graphs with your property can be characterised by forbidding as induced subgraph $\overline{P}_3$. By Andreas Blass's comment, your property implies that the complement of the graph is a disjoint union of cliques. These graphs are called complete multipartite graphs.

Sometimes for this kind of question the site http://graphclasses.org/ can be useful.

The second property is almost exactly the concept of "distinct subset sums", about which there are many beautiful unanswered questions. See, eg, http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums The only difference between what you ask and the classical case is that you allow for repeated values.