First Option is false because chromatic number 'k' not necessarily means that graph contains a clique on 'k' vertices. For example, Chromatic Number of a Cycle Graph on odd vertices is 3 but it doesn't contain any complete subgraph on 3 vertices.
Second Option is True by virtue of Pigeon-Hole Principle. If there are 'k' color classes and n vertices, then one color class size is atleast n/k. Note that, a color class is nothing but an independent set.
Third option is True because minimum number of edges between k color classes is C(K,2) or 'k choose 2'
Fourth option is False because take a cycle graph on odd vertices, chromatic number is 3 but degree is not 3 for any vertex. Similarly, for any complete graph, degree for every vertex is n-1 but chromatic number is n
