Yes. Such cliques are called rigid families of graphs. For every infinite cardinal $\kappa$ there exists a rigid family of graphs of cardinality $2^\kappa$, such that each graph in the family has $\kappa$ vertices.

This is classical so I was surprised by the trouble of finding a neat reference tailored to your question. An overkill reference is Theorem 1 in Section 4 of P. Hell On some strongly rigid families of graphs and the full embedding they induce. Algebra Universalis 4 (1974), 108–126

Yes. Such cliques are called rigid families of graphs. For every infinite cardinal $\kappa$ there exists a rigid family of graphs of cardinality $2^\kappa$, such that each graph in the family has $\kappa$ vertices.

This is classical so I was surprised by the trouble of finding a neat reference tailored to your question. An overkill reference is Theorem 1 in Section 4 of P. Hell On some strongly rigid families of graphs and the full embedding they induce. Algebra Universalis 4 (1974), 108–126

Edit: Actually rigid families of graphs require a stronger condition - no nonidentity homomorphisms (including endomorphisms) between its members. But of course every rigid family satisfies your requirements.